An immersion-oriented player is going to try his-her best to do exactly this, as that's the whole point of immersion: to perceive things as your PC would perceive them.
The PCs perceive exactly as the players & GM imagine they perceive, with the complication that they all need to be on the same page.
So, in one sense, perceiving as your character is easy (because their perceptions are under your conscious control), and, in another, impossible (because you can't un-know that).
TTRPGs - as opposed, say, to LARPs - present tremendous barriers to immersion by their very nature, overcoming them seems to be a rarefied, fleeting, and intensely personal thing.
Obviously. But that's just table knowledge.
So are the specifics of the system you're using.
No, but in theory they would have perceptions, knowledges and beliefs given that they are in theory sentient inhabitants of their setting;
Not on theory nor in fact, rather in our imaginations.
and those perceptions, knowledges and beliefs don't extend to seeing little tags on foreheads saying PC or NPC or BBEG or whatever.
Exactly. So, we don't (unless breaking the 4th wall) imagine that they're aware of their status within the game - even though you are necessarily aware of that status.
Put another way: imagine these characters are real people
That's redundant, they're necessarily imaginary people, by definition.
Now, is everything you just saw through all those characters' eyes consistent with itself no matter which set of eyes you happened to look through - whether it was a PC, an NPC, a commoner, a minion?
If that's how you imagined it, yes. If not, no.
Then, if no, you gotta ask yourself: "Self, why did you imagine it that way?"
But, if every persons perceptions of a hypothetical world were aligned and formed an internally consistent world, it'd be a very unrealistic setting. (Which is fine, fantasy should get a free pass on realism.)
bar full of common working people who look ready to fight, and yep: there they go. Fists flying, bottles smashing, a good old-fashioned donnybrook - black eyes all round and maybe a few broken bones, but in the end nobody dies and the bartender has a big mess to clean up.
Ok, so the bartender sees that as y'know, Friday. We see it as establishing a little something about the setting & genre - violence in the setting is common, but not quite as dangerous or acrimonious as in reality.
This time, however, most of the people involved drop dead the first time they take a good hit from anything - including getting hit by the same guy that hit him last week - because one of the new people has a PC tag on its head and suddenly all these brawlers are panes of thin glass.
I'm sorry, are you saying they see the tag, but, otherwise, he's just another brawler? Or that he, like the bartender is just watching the show?
Also, why has the lethality of violence shifted? Even in the presence of rules that speed up combat like that, non-lethal attacks presumably remain non-lethal, no?
How in any way is this internally consistent?
It's not, nor is it consistent with the use of minion rules & the presence (or participation) of a PC that far above the level of the crowds regularly scheduled violence, that they need to be modeled as minions.
Rather, if the PC in question just watches the fight, nothing changes: it's narrated just like the one from last week.
Similarly, if he's incognito, and trying to stay that way, the resolution won't be a combat, he might make some checks (depending on the system) to conceal his prowess, just put in a good showing to fit in.
If he does wade in full-bore, though, things change. There's a regular whirlwind of destruction all of a sudden, the new guy is knocking the toughest regulars cold as fast as they can come at him.
What do the regulars do? Team up and all teach the new guy a lesson? Or are they too preoccupied with their personal grudges and squabbles?
If the former, the whole fight gets played through and either the PC leaves the regulars unconscious or just plain given up (or fights his way out ofbthe place), or they bear him down by weight of numbers.
If the latter, the rest of the fight is just a backdrop, only the patrons that actually emerge from it to take on the PC get resolved under the minion rubric, the rest is narrated (and it doesn't matter if some of those who do take him on may already have battered in the narration of that backdrop of a barfight or not - not needing to track such things is part of the point, it speeds up the combat and reduces the bookkeeping burden on the DM, if it's also a little more abstract, well, the system it's appended too is likely pretty abstract, already).