Eldritch Strike is unbalanced
In an nutshell, Eldritch Strike allows a properly built warlock to benefit simultaneously from the invocations that improve weapon attacks and invocations that improve Eldritch Blast.
Moreover, it lets the warlock harness more damage from Hex and Hexblade's Curse.
This all adds up to a significant boost in power.
Eldritch Strike comes with problems, some of which you mentioned in your balance assessment, but these problems are either minor, avoidable, or both:
- Eldritch Strike requires a free hand, so no shields.
- This problem is generally true for most melee warlocks anyway.
- Melee warlocks often get War Caster which basically solves this problem.
- Eldritch Strike requires the attack to hit, so it can't be used when no attacks hit.
- This invocation is mostly of interest for Pact of the Blade warlocks who get a second attack via Thirsting Blade, which greatly mitigates this problem.
- Eldritch Strike does not function with two handed weapons.
- This problem can't be worked around, but most of the warlock's damage comes from other bonuses, so the size of the weapon die is not a big deal.
- In most scenarios Eldritch Blast will be at a disadvantage if the warlock does not want to move back and trigger an opportunity attack.
- Enemies with a 10 foot (or more) reach do not cause this problem as the warlock has some leeway to move back without provoking opportunity attacks.
- The Crossbow Expert feat directly solves the problem.
- The Mobile feat solves this problem indirectly by letting the warlock move back without provoking opportunity attacks.
- Spells like Darkness or Shadow of Moil similarly solve this problem indirectly because unseen creatures do not provoke opportunity attacks.
- One handed melee weapons with the thrown or reach property, such as daggers or whips also mitigate this problem because the warlock can occasionally make an attack from 10 or more feet away. (Note: "attack with a melee weapon" is not the same as a "melee weapon attack")
- The Repelling Blast invocation also mitigates this problem at later levels, because if any of the beams hit, the enemy is pushed and the rest of the beams are no longer at disadvantage.
For a rough comparison, consider the Bonus actions granted by Two Weapon Fighting, Polearm Master, or Maddening Hex. Eldritch Blast becomes way stronger than any of those Bonus Actions once it gets multiple beams and Agonizing Blast.
Another rough comparison is Sorlocks as they can also cast Eldritch Blast as a bonus action. However, they must spend 2 sorcery points to do so, whereas Eldritch Strike is at will.
If I were to build a warlock to make the most out of Eldritch Strike I would create a Pact of the Blade Hexblade with Mobile, Agonizing Blast, Eldritch Strike, Lifedrinker, and Thirsting Blade.
In the endgame this warlock deals (1d8 + cha*2)*2 + (1d10 + cha)*4 = 71 at will damage per turn, but the warlock can deal another 6*6 from Hexblade's Curse and 6d6 from Hex for a total of 128 damage.
Roughly 1 in 5 turns the warlock can't cast Eldritch Blast due to missing both melee attacks, so the adjusted damage is 64 at will and 112 with resources.
Just for reference, a typical melee Pact of the Blade Hexblade with a glaive, Polarm Master, and the relevant invocations deals (1d10 + cha*2)*2 + (1d4 + cha*2) = 43.5 at will damage per turn or 72 with resources.
The warlock might occasionally get some extra reaction damage from the Polearm Master feat, but it won't make up the difference.
The intent is that you can only damage one cursed target (and creatures next to it) at a time
The invocation repeatedly refers to a single cursed target
As you note, the Maddening Hex eldritch invocation says (XGtE, p. 57; emphasis mine):
As a bonus action, you cause a psychic disturbance around the
target cursed by your hex spell or by a warlock feature of yours
[...] When you do so, you deal psychic damage to the cursed target
and each creature of your choice that you can see within 5 feet of it.
[...] To use this invocation, you must be able to see the cursed
target, and it must be within 30 feet of you.
The feature doesn't explicitly say the damage is dealt to only one cursed target, but it repeatedly refers to the target in the singular (i.e. "the [cursed] target", or "it"). This suggests that even if you do have multiple curses active - such as a hex on one creature, and a Hexblade's Curse on another - you simply choose one of these cursed targets when you activate Maddening Hex, dealing the damage to only that creature and other creatures (of your choice) adjacent to it.
Designer intent indicates that Maddening Hex triggers off just one of the creatures you have cursed
Rules designer Jeremy Crawford answered this exact question in an unofficial tweet from November 2017:
For Maddening Hex, does the psychic damage trigger off of every target cursed by you within range, (One creature cursed by Hexblade's Curse and another cursed by the spell Hex, or the like) or only a single cursed target per turn?
Maddening Hex works on one cursed target at a time.
Crawford's unofficial ruling seems to match the wording of the feature. You may have multiple targets cursed at a time using different warlock features and/or the hex spell, but Maddening Hex only activates on one such target. As you point out, it says "the [cursed] target", singular - so even if you've cursed multiple targets, you can only use Maddening Hex to deal damage to one cursed target (and creatures adjacent to that one, of your choice) at a time.
Best Answer
There is no spell attack associated with hex. You cast it on the target and it affects the target until you stop concentrating or until the duration ends.
Hex doesn't cause any direct damage to the target; it adds an additional 1d6 to other attacks you make against the target. If the target dies you can move hex to a new target.
Note that this is different than most spells that don't require a spell attack roll, in that it always succeeds. For hex, there is no save to resist the spell or diminish the effect as there is for other non-spell attack spells which have negative effects for the target(s).
This spell is similar to the ranger spell hunter's mark, which has the same behavior.