Levelling and equipment progression, especially vial loot are not really specifically shooter mechanics, or, at least, I wouldn't say that. Plenty of shooters - at the very least the 'boomer' ones - do not have those, thankfully.

RPG elements seem to be an addictive hook for many people. I think that in most cases they are a very shallow addition to a game - you basically just 'progress' for the gameplay to largely stay the same, or, worse, for the gameplay to become even more and more one-note.
The case of 'basically no progression' happens in games where you will mostly, if not always, be in places where your level and your equipment are guaranteed or near-guaranteed to be appropriate. You will deal with encounters in roughly the same ways, all the time, and the complexity of your choices will largely stay the same since some point where you get your crucial tools for dealing with them. The name of some of your tools might change, but that's kind of it.
The case of 'regressive progression' happens when either some of the tools that you have become strong enough to dominate encounters, and, perhaps, with the other tools becoming too weak. This makes it so that your choice of tools becomes simpler and simpler as the game progresses.

Oh, yeah, one of the things that also breaks verisimilitude for me rather often is health, and this also affects tabletop RPGs for me. Characters gaining health with levels very often means that they can somehow deal with situations that I don't think experience can help you this much. The classic case would be D&D characters with their roughly linear health growth, and how two characters of different levels falling from the same height would very likely get different results. Or consider how burning and bleeding are basically non-factors in BG3 (and, in at least the case of burning ground, - in DOS2) at higher levels. Massive health growth with levels very often just hurts my enjoyment of things.

I do like these RPG elements in at least some cases, though. At the very least in XCOM and similar games, where that progression is actually meaningful, because you can, and will likely lose experienced soldiers, making it so that you do not always 'progress' in the sense of RPG mechanics, but also go back to your weaker state.