No, that would be immersion breaking.
The general point is, if you want more historical simulation then play a game that is a historical simulation. EU is that, Civ is not. Civ uses history as a theme, but at no point does it ever try to be an accurate presentation of actual history. So its like getting mad at a screwdriver for not being a good hammer.
EU is much less focused on city-building and doesn't allow you to enjoy the progress of your civilization on a micro level as much as Civ does. You're proposing to replace a screwdriver with a saw - still not the hammer that's needed for the job.
There is certainly an itch that neither EU nor Civ scratches as fully as some players desire - a history simulation game that is more focused on actual city-building and allows you to see how your nation grows from the humble beginnings of a small tribe. Civ 5 tapped into this realm by looking more realistic than its predecessors, and I think it attracted quite a number of players that have been craving for this particular niche. But now, with Civ 6 gong back more to its "board game" style, we can see the divide in what players seek from the franchise.
More on topic: it looks like there are several issues with the governor system when players claim that it's immersion-breaking, and lumping them all together seems to just add confusion to the discussion:
1. Governors not being from your civ. I personally have no issue with this one. We don't yet know when you can earn your first governor (or do we?) If it becomes available starting from Medieval era, then I'm not that concerned - by that time there were plenty of real life nations that had foreigners in their courts in one capacity or another.
2. Governors being from civs not in the game. This is a grey area, and really up to one's imagination and limits of immersion. I personally can turn a blind eye and assume they are coming from City State migrants of similar cultures. IMO, this aspect in particular is the one where you can't straight out reject it while accepting the fact that your Great People, and especially GWAMs, are all over the place.
3. Same governors/personalities being available to all civs. I can see this one being somewhat silly, and it could've been an easy fix. In fact, I think it would've been easier for them to just name governors as "The Castellan", "The Surveyor", etc. instead of having to come up with names. They clearly wanted to give the governors a bit more personality without having to come up with art assets every time they come up with a new civ. Maybe they do think of governors as spiritual successors of advisors, but didn't consider the shoft in the role's significance.
4. They look waaay too cartoony. Guess we'll have to swallow it... although I can see them growing on you eventually. Except for Pingala - you can't just remove Teddy's cheeks and stitch it to another guy...