[RPG] Are Eldritch Knight cantrip choices limited

While the question you ask is theoretically simple, there are a lot of factors that need to be taken into account when dealing with this. The difference between the two builds isn't just "Oh, I'm going to change out a couple of levels and have basically the same character!" There's a lot of differences, and there can be a lot of things that change between the characters. For simplicity, all of the characters will share the same race and ability scores. This means that I can calculate the same attribute bonuses how I like, and basically make the character have the skills we need.

The first decision is to assume we are going a strength-based character. We could do a bow Fighter, and it would work well especially with the long-range of the bow, but I am still trying to max out the Champion aspect of the Fighter/Wizard, and using a Str-Based character with a Greatsword means that the critrange of 19-20 is helped out that much more by 2d6 melee over 1d8 ranged damage from a Longbow.

The goal for the Knight is to get to basically max Str/Int for damage purposes, and then either 18 Con + Tough for max HP, or 20 Con for 25 less HP than the tough mode and gain +1 Con Mod.

For the Fighter/Wizard, we want to maximize damage. While this class should still have a fair bit of melee ability, we need to maximize the offensive abilities best we can. Trying to max out Str/Int first is good, and then having the rest of the ability score increases (which there are only 5 of) to round out the HP gives the Fighter/Wizard a significantly worse HP.

We are going to use a Mountain Dwarf. Since we only need Int, Str, and Con as our stats to max now, since Wis, Cha, and Dex don't matter, we want to maximize those values. Mountain Dwarf gives +2 to Str and +2 to Con, so we can start out with a 17/8/17/15/8/8 character. With the Eldritch Knight, it's impossibly easy to get 11 points of Attributes out of the 7 attribute increases.

Itemization will be the same. They will both have 2 +1 Longswords/Shortswords and +1 Full Plate (for defensive purposes). This will give them max AC and max damage for now.

The target will be an Ancient Red Dragon. It has 22 AC, which to my knowledge is the hardest thing to hit at the moment.

From there, it becomes a simple measure of planning and building the characters.

Elder Rich Level 20 Mountain Dwarf Fighter (Eldrith Knight)

• Max HP: 244 (10 + 19*6 + 20 * 4 + 20 * 2)
• Proficiency: +6
• Strength: 20 (+5)
• Dexterity: 8 (-1)
• Constitution: 18 (+4)
• Intelligence: 20 (+5)
• Wisdom: 8 (-1)

• Charisma: 8 (-1)

• Start: 17/8/17/15/8/8

• 4th: Dual Wielding
• 6th: Heavy Armor Master (Str to 18 (18/8/17/15/8/8))
• 8th: Tough
• 12th: Str to 20 (20/8/17/15/8/8)
• 14th: Con to 18, Int to 16 (20/8/18/16/8/8)
• 16th: Int to 18 (20/8/18/18/8/8)
• 19th: Int to 20 (20/8/18/20/8/8)

Features: Two Weapon Fighting (Fighting Style), Second Wind, Action Surge (Two Uses), Weapon Bond, Extra Attack (3), Indomitable (three uses), Weapon Bond (Eldritch Knight), Improved War Magic (Eldritch Knight), Eldritch Strike (Eldritch Knight), Arcane Charge (Eldritch Knight)

• Spell Slots: 4/3/3/1

• Feats: Tough, Dual Wielding, Heavy Armor Master

• Main Hand: +1 Longsword (+12 Attack, 1d8 + 12 Damage (16.5 Avg. Dmg.), 2d8 + 12 Crit (21 Avg. Dmg. Crit))

• Off Hand: +1 Longsword (+12 Attack, 1d8 + 6 Damage (10.5 Avg. Dmg.), 2d8 + 6 Crit (15 Avg. Dmg. Crit)) AC: 20 (+1 Full Plate, Dual Wielding)

Rich is probably going to be able to deal a fair bit of damage. Each single Main Hand Longsword strike will deal an average of 16.5 Damage on a hit. Against a Red Dragon, that's a 55% chance to hit (and 5% to crit), so 9.3 damage per swing accounting for accuracy (and crit). The offhand is 10.5 damage on average, so a 6 damage per swing accounting for accuracy (and crit). That means every attack action is going to be 15.3 damage per swing. Considering that Rich will get 4 attacks per turn, this is 61.2 DPR on just a basic set of attacks. With an Action Surge, that damage jumps to 122.4, which is approximately 22% of an Ancient Red Dragon's max HP. Two turns right after each other, Rich Alone could easily bloody the Dragon with slightly lucky rolls.

The main hand weapon used to be a +1 Greatsword. Compared to the current build, it did less damage on Rich's turn (10.8 damage per Attack action, 43.2 damage per turn vs. 15.3 per Attack Action, 61.2 per turn), but had a higher reaction -> opportunity attack damage. This is because it had an average damage of 19 Damage (Crit 26). These extra points of damage could stack up well, but to be honest, the lack of 2.5-5 points of damage on the 1 reaction per round loses out to the heavily increase DPR thanks to the dual wielding. The Greatsword also had a higher Improved War Magic damage, thanks to being able to attack only with your main-hand weapon. Again, while the loss of 2-5 damage per Spell -> Attack is known, the heavy increase in overall DPR from the multiattack functionality of Fighters is much more useful.

Fig Tier Wiz Level 20 Mountain Dwarf Fighter (10)/Wizard (10)

• Max HP: 184 (10 + 9*6 + 10*4 + 20*4)
• Proficiency: +6
• Strength: 20 (+5)
• Dexterity: 10 (+1)
• Constitution: 18 (+4)
• Intelligence: 20 (+5)
• Wisdom: 8 (-1)

• Charisma: 8 (-1)

• Start: 17/8/17/15/8/8

• 4th: Heavy Armor Master (Str to 18 (18/8/17/15/8/8))
• 6th: Str to 20 (20/8/17/15/8/8)
• 8th: Int to 17 (20/8/17/17/8/8)
• 14th: Int to 19 (20/8/17/19/8/8)
• 18th: Con to 18, Int to 20 (20/8/18/20/8/8)

Features: , Second Wind, Action Surge (One Use), Improved Critical, Extra Attack, Remarkable Athlete, Indomitable (one use), Great Weapon Fighting (Additional Fighting Style); Arcane Recovery, Arcane Tradition, X Savant, 3 Arcane Tradition Features (Arcane Ward, Projected War, Improved Abjuration || Sculpt Spell, Potent Cantrip, Empwered Evocation)

• Spell Slots: 4/3/3/3/2
• Feats: Heavy Armor Master

OPTION 1:

• Main Hand: +1 Greatsword (+12 Attack, 2d6 + 12 Damage (19 Avg. Dmg.), 4d6 + 12 Crit (26 Avg. Dmg. Crit))
• Feature: Defense (Fighting Style)
• AC: 20 (+1 Full Plate, Defense)

OR

OPTION 2:

• Main Hand: +1 Shortsword (+12 Attack, 1d6 + 12 Damage (15.5 Avg. Dmg), 2d6 + 12 Crit (19 Avg. Dmg. Crit))
• Off Hand: +1 Shortsword (+12 Attack, 1d6 + 6 Damage (9.5 Avg. Dmg), 2d6 + 6 Crit (13 Avg. Dmg. Crit))
• Feature: Two Weapon Fighting (Fighting Style)
• AC: 19 (+1 Full Plate)

So here I've given out 2 builds for Fig that both can work. We only have 5 Ability Score increases, so it is much more difficult to max out Str and Int and get a high Con score. Technically, Heavy Armor Master can be exchanged for any feat that gives +1 Str, Con, or Int, but the DR 3 on non-magic weapons seems like a huge deal.

Option 1 is what I had originally come up with for Fig and Rich. The idea is to max out damage thanks to Str and Int, and use a Greatsword for combat. Because of that, you can take the Defense Fighting Style for +1 AC. The damage for this character is 10.8 Damage per attack accounting for accuracy against an Ancient red Dragon, and therefore a 43.2 Damage per turn.

Option 2 is more like Rich, wanting to use two weapons. The best/only options for max 1-handed light damage is Shortsword basically. As with Rich, the damage is greater over the course of a round than using the Greatsword because of the multiple hits you get in per attack, but you have a lower amount of damage on a reaction, by about 2 points (3.5 points on a hit, 7 points on a crit). This build sacrifices 1 AC to get an increase of damage at close range.

(As an aside, Option 2 could also possible use an 18/8/18/20/8/8 final ability score with 2 +1 Longswords and Dual Wielding. I checked the numbers, and while you do get +1 AC thanks to dual weilding, you actually lose damage overall. This is because while you do get to go from 1d6 weapons to 1d8, you lose +1 to your Attack Modifier and +1 to your damage. The mainhand would be +11 to attack, 1d8 + 11 Damage (15.5 Avg. Dmg), 2d8 + 11 Crit (20 Avg. Dmg. Crit)). As you can see, the main hand would basically lose 5% on normal attacks, while the Average Crit damage increases by 1. This is the same for the off hand. The only difference is basically that you lose 5% to-hit to gain a marginal benefit when you crit.)

So right off the bat we see that Fig is lacking in a couple of areas compared to Rich. He's already at a much lower HP thanks to the levels in Wizard only giving +4 to HP per level instead of +6. In addition, the lack of ability score increases means that hitting 18 Con + Tough or 20 Con is almost impossible without sacrificing damage. With 60 HP less than Rich, it's a fair bit of damage that is not turned lethal.

Second, the lack of Action Surges and Extra Attacks means that Fig won't be able to attack more than twice a turn. This is huge when you compare the two characters. The lack of Dual Weilding for Fig means no "natural" dual wield longsword, and therefore a fair bit of loss of damage. On top of that, you can only deal ~28.2 damage per turn, ~56.4 with your single Action Surge. The damage that Fig is going to be putting out is on the level of a 10th level fighter, not a 20th Level Fighter/Wizard.

Speaking of the Wizard side, you are still not a super-powerful spellcaster when it comes to spells. What you give up from the Wizard Side to "splash" 10 levels in Figher means that you only have access to 5th level spells. While not an issue for the most part, 5th levels spells will need to greatly make up for the damage deficit that the lack of extra attacks the fighter should have incurs. You also "only" have 2 5th level slots and 2 4th level slots over the Knight. I put "only" in quotes because while I can attempt to downplay those slots, they represent a level of power the Knight cannot achieve. If we are talking about pure damage, the Knight can cast 1 4th level spell, but will probably want to prefer to use his spells as support spells to further lock-down his targets while he beats the heck out of them. The Wizard can use those 4 extra slots to provide a lot of utility to the party.

For comparison of the spell casting abilities, we need to look at the damage between the classes. Because spell damage is only based onthe spell, not your For now, I think the best option to check for spell damage is going to be the Spell Damage Comparison Chart put out by Rhaegar83 on /r/dndnext.

Looking at the spells, it's clear that no 5th level spell can match the single-turn damage of a 20th Level Eldritch Knight attacking the crap out of a single target. While you do get access to some nice 5th level AoE spells like Cone of Cold and Flame Strike, the Eldritch Knight can easily hit 2 targets for similar Damage if he is able to get between the two using his move.

The end result is that Rich, the Eldritch Knight is going to end up tankier (HP and AC) and able to deal more single-target damage thanks to his quad-attack. Fig is going to be able to deal more AoE damage, single by virtue of having access to 5th level spells, as well as 2 more 4th level spells slots. Fig is also going to be able to be more of a controlling presence on the battlefield, if he is going to take at least some support/control-y spells.

You can switch out any of the "any-school" spells

The PHB Errata, has now fully solved this issue.

From the "Rogue" section of the errata:

[New] Spells Known of 1st-Level and Higher (p. 98). In the final sentence of this subsection, the closing phrase has changed to read as follows: “… unless you’re replacing the spell you gained at 3rd, 8th, 14th, or 20th level from any school of magic.”

Similarly, the "Fighter" section of the errata states:

Spells Known of 1st-Level and Higher (p. 75). The final sentence ends as follows: “… unless you’re replacing the spell you gained at 3rd, 8th, 14th, or 20th level from any school of magic”.

This means you can indeed retrain the "any-school" spell and replace it with another "any-school" spell as appropriate, both as an Eldritch Knight and an Arcane Trickster.

As a historical note, the fighter section was added to errata at least as early as 2016 while the rogue section was not fixed until 2020 though this tweet from Jeremy Crawford already stated that the same reasoning should apply:

Like an Eldritch Knight, an Arcane Trickster can retrain their 1st-level spell from any school. (An unintended omission.)