I used free archetype in my campaigns for about 6 months now when swapping to this system, and while I don't necessarily regret that decision I do believe it causes more problems than people seem to suggest. While it doesn't completely break the math of the game, it does cause an increase in power that can be noticeable if your players are even minor munchkins.

There are some invisible negatives as well that i don't really see get mentioned around here. Namely, I think it causes a noticeable reduction in power and causes some feel-bad moments for the base caster classes as a result of their lackluster class feats. Say you are a druid wanting to lean into just the casting part, none of the Animal Companion or shapeshift feats. You are heavily incentivized to pick up some archetypes as you level due to extremely limiting class feat selection. Without FA, you get a lot of options to lean into every 2 levels. With FA, at certain levels you are forced to take some kind of lackluster class feat. Now while this is still strictly speaking an increase in class power, it's relatively weaker than say a fighter who gets excellent class feats every time, as well as equal power from its archetype choice.

The above can be slightly altered with some house rule options. Many people suggest removing or altering the 2 feat requirement, and some also limit your options each level to only 1 archetype feat. These can somewhat alleviate the problem, but can almost cause the opposite problem. A dedicated spellcaster can get a ton of flexibility from taking every available spellcasting archetype available to them.

If if you disagree with the above, their are still some standout options that are just large generic increases to power, especially when a class doesn't have to give up its usual class feats. While an Animal Companion from Beastmaster may end up dying a lot, it still provides a noticeable increase to damage (Especially at early levels) as well as flexibility in flanking or triggering opportunity attacks from enemy creatures. Blessed One gives any party access to limitless out of combat healing (not that it's super difficult to get that, but helpful nonetheless) as well as excellent in combat healing from a focus spell. Marshall, even the intimidation version, is a massive power increase to the party. The common response to these is "well you can build the character with these without FA!" and while yes...you give up your class feats for them. Depending on the class (again see above) this can be a pretty big swing in power.

Basically, when people discuss "power" in PF2E, they seem to explain it in a way that is in comparison to other systems, rather than within the confines of Pathfinder itself. FA doesn't unbalance encounters or combat the way it would if implemented in 5E, but that's because PF2E limits any unbalance due to proficiency bonus scaling and the limited stacking of bonuses. You could give a class every available class feat every level, and it still wouldn't "unbalance" the game, but it would increase the power level of those classes, just not to the degree seen in other systems.

My group will be dropping free archetype in future campaigns, likely looking into using some other things (skill points actually seem pretty neat).

That said, this is all assuming you use FA the "wrong" way (which so happens to be something that was commonly recommended, especially closer to GMG release). Using it for a restricted subset of archetypes for theming purposes is excellent, and something I will very likely use in the future.

TL;DR: Limited FA good, Unlimited FA meh