In terms of the Balance of the class, I'd say you don't have a ton to worry about. Artificers only gain access to spells up to spell-level 4, and even then they only gain access to those spells around level 19 anyways, so regardless of which lists they can pull spells from, they're not going to get tremendously powerful magicks irrespective of their level.
There is one important consideration though: by vanilla, Artificers don't gain access to Evocation Spells. The only two evocation spells they get are Cure Wounds, a healing spell, and Continual Flame, a non-damaging spell. In fact, if you iterate over their list of spells, you'll see that there isn't a single spell they can cast that directly deals damage to anything. The closest is Mordenkainen's Faithful Hound, which summons a spectral dog which can attack enemies.
Meanwhile, Artificers do get spells like Haste, a very powerful spell that greatly improves a creature's damage potential, but which doesn't itself directly deal damage. All of the spells they get are utility in nature: healing, ability improvements, invisibility, flight, etc. Now, I can't speak to designer intent, but my sense is that this is a deliberate choice, that Artificers aren't meant to be Offensive Spellcasters.
Your change, which would effectively be allowing Artificers to gain access to offensive spells, would have pretty significant implications on the balance of the class. I hesitate to suggest it would make them overpowered at all, because A) they'd get a lot of the really powerful low-level spells way after their prime, and long after they'd be tide-turning, and B) Artificers are generally regarded as being underpowered in the first place.
But it would change how they play, and how they treat their spellcasting abilities. My general advice, then, is that you shouldn't change their spell list.
I do like the ability of giving them some cantrips, though I'd advise you continue the practice established by their spell list, and only allow them to pick non-damaging cantrips. Note that stuff like True Strike would probably be permitted, since it doesn't directly deal damage, and fits the theme of spells like Haste, which is already in the Artificer Spell List.
You cannot use the wand more than once per turn with these two features
Using a wand is a special action defined by the wand's description. The prototype wand says:
As an action, you can cause the magic wand to produce the cantrip [...]
You cannot produce a cantrip from a wand with the Attack action.
When you take the Attack action, Arcane Armament allows you to attack two times in that one action. Attack meaning to attempt to hit something with a weapon. But you cannot replace an attack with another action. And using the wand to produce a cantrip is not considered, in and of itself, to be an attack. Cantrips may have attacks as part of their effects, but producing a cantrip is not itself an attack.
When you produce a cantrip from the wand you have to use your full action to do so, which means normally giving up the option to Attack.
So, you cannot take an attack given to you by the Attack action and use it to use the wand to produce a cantrip.
Best Answer
Usually, Not Much
As you've realized, the artificer class has somewhat strange requirements that seem more specific than necessary. At low levels, it really doesn't matter whether you're using artisan's tools or a normal focus. At high levels, you have a lot of options that are superficially similar with minor differences. These can probably be reasonably flattened with little to no mechanical impact.
...Unless You're Multiclassing
This is really the main balance implication that I'm aware of. Having played specifically an Artificer/Wizard multiclass build, I can also say that it's not exactly a small one.
You require artisan's tools to cast your artificer spells, but those cannot be used to replace the material components for your wizard spells. Further, the spellcasting foci gained by the artificer subclasses are also restricted to artificer spells (and I would argue that the infused items implicitly are as well, since they're replacing the tools).
This conflict imposes real costs on certain Artificer/Wizard multiclass decisions, especially one which wants to dedicate one hand to a shield. You're pigeonholed into finding something that can work for both. Enhanced Arcane Focus is great for this, but it uses up an infusion slot, and you will likely come across things you'd rather be holding with that hand.
Otherwise, you're left with doing complicated action economy things like sheathing your tools one turn to use a component pouch, and hoping you won't need to cast a spell as a reaction before you can draw them again.
...But Even Then Only Sometimes.
This is all very frustrating, until you get to artificer level 5. Now suddenly even magical arcane foci are on the table, and you have your infusion slot back.
But is getting to artificer level 5 desirable? For my purposes, a three level dip gave me everything I wanted from the class, while keeping eventual access to 9th level spells in a campaign that would someday hit 20th level. Pushing to level 5 for just this convenience is also not a trivial decision.
So, Maybe. Sometimes.
This is the only place I'm aware of where there is a severe mechanical impact to the distinction. Like so many things in D&D, it's extremely contextual. But in my circumstances, your simplifying ruling would effectively give my character an additional infusion slot back, and the ability to gain stronger spellcasting foci indefinitely.
Remember, this isn't a hypothetical. This is a particular character build that does exist, that would gain substantial mechanical benefit from your ruling.
Does that matter to you? Honestly, probably not. Dungeons & Dragons has so many contextual mechanical interactions that even if a rules choice unbalances some of them, it's extremely easy to just never encounter them in a campaign. If you aren't multiclassing, you're probably fine. Even if you are multiclassing, you're probably fine.
But artificers are a complicated class, with a lot of options, and giving them more options does have consequences. For someone. Somewhere.