It will be fine for Rangers

Rangers are arguably the weakest class in the game. It has virtually no chance to be overpowered, but could end up underpowered.

Is it overpowered?

The strongest combination I was able to find for level 20 is this:

• Gloom Stalker 11, Stalker's Flurry: Reroll one attack roll
• Gloom Stalker 3, Dread Ambusher for an extra attack on the first turn
• Hunter 3, Colossus Slayer for 1d8 damage on one of the attacks
• Horizon Walker 3, Planar Warrior for 1d8 extra damage on one of the attacks
• Monster Slayer 3, Slayer's Prey for 1d6 extra damage on one of the attacks

So in the first turn he makes 3 attacks (Extra Attack + Dread Ambusher) +1d8 + 1d6 damage, and 2 attacks for the rest of the encounter with 2d8 + 1d6. He also rerolls one attack. Except for the first turn this is less than the 11th level Fighter without any archetypes. If we add that, it easily compensates for the first turn.

Is it underpowered?

If he does not select Beastmaster as one of his archetypes, he will be a valid character.
Beastmaster is the weakest archetype, delaying its features further could make him seriously behind other characters.

Other classes

It all comes down to how powerful the archetype features are, and how early you get them. You have to examine every combination.
If you reach the level where the final archetype feature normally comes you should probably stop, as any future features would make you stronger than any of the single archetypes. But as the Ranger above shows, some classes are weak enough to cross even this limitation without getting overpowered.

Clerics for example get great things from their domains even on level one, so for them this would be definitely overpowered.
Bards on the other hand would probably become weaker without sticking to any one archetype until level 14. After that any new archetype features would be net gain, because you get the last archetype feature on level 14 normally.

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.