This is a Massive Change
While it might seem like a small change, this is actually a massive change. There are numerous effects that end at the start of a turn or end of a turn that can be massively affected by initiative order, and some abilities will be outright overpowered unless also house ruled. While an exhaustive list of effects would be probably hundreds of pages, I'll just list a few significant ones here:
Initiative Feats and Abilities
Some feats and abilities trigger "when you roll initiative". A safe bet to house rule all of these is to limit them to the first round of combat, but unless otherwise modified, there are some potentially overly powerful feats. These include Swaggering Initiative (free weapon draw once/round instead of once/fight), After You (panache EVERY ROUND for simply going last), and the Triumph Siktempora's Uncanny Pounce (two free actions every round). Really, there are many of these abilities, but basically all of them should be sane if limited to just the first round.
Certain effects last until the "start/end of your next turn", or the same but for the target's turn. These can be drastically reduced in effectiveness depending on your relative initiative positions. Take the ever common Raise A Shield action. If you are last in a round, and spend an action to raise a shield, then roll the highest in the next round... you just wasted your action. Conversely, if you are first in a round, raise your shield, then last in the next, you just got two rounds worth of value for that one raise a shield action. This applies to other actions, like Inspire Courage, or Guidance, even making things like Aid potentially unreliable if your ally doesn't get a turn before your next turn. There's not going to be a consistent rule to re-balance all of these - instead, creatures with abilities like these will only be incentivized to use them if they roll high initiative, and dis-incentivized if they roll low. This will lead to more meta-strategy: strategic decisions your character makes because of the rules of the game instead of in-world situations.
Frightened and Persistent Damage
Let's say I am second to last. After me is the monster, so naturally I Demoralize it. I critically succeed! The monster then goes, reducing it's frightened value to 1 at the end of its turn. Then rolls top initiative, reducing it to 0 at the end of its turn... my allies had no opportunity to capitalize on that! Conversely, if I have a Wounding rune (or other means of reliably applying persistent damage), then if I benefit less from going twice in a row than others would, since persistent damage doesn't stack. However, I get more benefit from the monster going twice before me than another would, since it's more likely I'll be able to re-apply my persistent damage. There's no house rule I can think of to fix these interactions - these again will increase the amount of meta-strategy, and increase the amount of unpredictability in general.
Stealth - Constantly Sneaking?
If you roll Stealth for initiative past the first round, do you get to become hidden if you beat Perception DCs? If not, why are you rolling Stealth anyway? This rule would need changed in some way.
If a monster takes out a PC, and that monster is last in initiative, with your current rule the PC would have to immediately make a stabilization check. This can result in a lot of deaths that would otherwise be react-able by the players. For example, if you go down to a crit (a common occurrence), then critically fail your recovery check, or are any level of doomed... you're toast. The remedy for this is less clear - I'd suggest that the players makes their recovery check starting at the end of the next round. This guarantees everything a chance to react to the dying player, sometimes multiple times, and makes the dying player more likely to miss turns. But it's easyish to remember and better than dying!
Swingy-ness, aka How The Party TPK'd
The biggest potential, non-mitigatable downside is that this makes the game more swingy as a whole. It's entirely possible for you to not get a turn between two of the big bad's turns. It's even possible for no one in your party to have a turn between two of the big bad's turns. Given how big bad's tend to be stronger than any individual PC (averaging at least 1 level above the party in my experience, to sometimes as high as 4), two turns can wreck a party. A dragon that breathe weapons, makes a no-map attack (three actions), then on the next turn does a draconic frenzy can easily take multiple party members down to 0 HP. Randomness like this will only hurt the players in the long run - it's more opportunities for things to go wrong in a way that they can't react to. That's my biggest worry about this house rule - enemies having back-to-back turns prevents the players from being able to react, and there's nothing less exciting than watching your characters die without being able to do anything at all about it.
Is This a Bad Rule?
Given the amount of things discussed above (which is just a subset), you might be tempted to think this is a unworkable house rule. However, if you accept meta-strategy as a fun and acceptable piece of the game and avoid uber-bosses (i.e. level+3 or higher enemies), it should mostly work out okay! It creates more randomness - some players will enjoy that, some will hate it. It's up to the GM to know their players.
Suggested Accompanying House-rules
If I were to run a game with the house-rules you described, I would add some more house-rules to maintain balance. I would add the following rules:
- Any feat that has you take additional actions based on "rolling initiative" only provides those actions in the first round. Alternatively, all feats that modify initiative only work the first round.
- Only Perception for initiative after the first round. This makes Perception vastly more important than it already is, but it makes the most narrative sense, and requires the fewest rule changes to make Stealth initiatives work.
- Recovery checks start at the end of the round after the round in which the player went down.