If a single enemy finds a stealthed party member, then I'd argue that only that enemy is automatically aware of the party. Other nearby enemies would still need to succeed their Perception check or be Surprised, making it very unlikely that you'd have situation where every single enemy can attack that PC before that PC (or the rest of the party) gets a turn.
I think I basically agree with the rest of your post, but on this particular point I’ve always thought of this as being a case of one enemy spotting us and sounding an alarm to their mates. Admittedly it would be nice to have each enemy type have a bark to reflect this, but I’m not in principle against groups of enemies getting the benefit of one of them spotting one of my characters, so none of them are surprised.
As long as all my party, including the stealthed ones, actually went into turn-based mode and got a chance to roll initiative to join in, that would be good enough for me. If my other party members were too far away to help in time then that’s my fault for not planning better. And if they are all just unlucky with initiative and my originally spotted party member takes a hammering as a result then them’s the breaks! At least it’s not as likely to happen as currently, and not a whole lot worse than what can happen in any battle.
I can see this might be a bit weird and end up in some jumping around and skipping turns if some of your party were a long way from the others, and you could even end up having two separate battles concurrently, with their turns mixed together, but I still think it would be better than the current setup.