The inherent problem is that of POV angle. First person may have seemed very immersive in the dawn of 3d games when the alternative was isometric. But no sane person will say that limiting his field of vision makes the experience more "immersive", in a modern narrative-heavy interactive movie-game.
It depends on the game. First-person has rarely been used in films just because it's incredibly restricted and, up until just recently when cameras were small enough to just strap to someone's head, was really difficult to pull off technically. Video games run into the same problems with first-person storytelling that films run into: it can be disorienting, it's difficult to identify with a character if we can't see them reacting to things, and it can be plain hard to convey some narrative details that are much easier to show via 3rd person (like, for example, things hidden from the POV character, stuff characters are doing off screen, and etc). There have been a few creative solutions to this in film and TV (Peep Show, for example, was entirely first person, but it would jump from character to character, so it wasn't restricted to just one person). However, I think video games, as a medium, are much more suited to fully FP storytelling than films.
Personally, I think its POV perspective is integral enough to the game's themes that it would hurt Cyberpunk to jump to 3rd person at any point in its narrative. Driving I don't mind, as it's mostly just a concession to the fact that it's much easier to navigate in a car in 3rd person. However, beyond that, Cyberpunk is a game about blurred identities and the threat of dissolution of the self allowed by the game's technologies, whether they are the relic that allows for Johnny to exist in V's brain, or Braindance technologies, or the slow and gradual replacement of the human body with technology (one recalls the Wizard of Oz's tin man's story about how he started replacing his parts with tin as he accidentally hacked them off until, one day, he had hacked all his human bits off and there was nothing but tin left). The game very effectively uses its first person perspective at various points to underscore these themes, especially in the bits that blur Johnny/V together to the point that you aren't immediately sure which one you are playing. Given that, it's kind of fitting that the player character rarely sees themselves anywhere in the game world, only at key points when they actively look into a mirror. Some of this might have been some weird limitation of the engine (the lack of mirrors, probably, although it's definitely strange that in a game so replete with beautiful reflections that seem to easily reflect everything in great detail, there are almost no places where the player can see themselves), but I think the decision to stick to first person in all other areas was intentional.
Kingdom Come wasn't a game about post-humanism, so I didn't really care or even notice when it shifted to third person for dialog. However, just switching to cinematic third person cut scenes at key moments was a laziest and easiest solution to the problem. The best, most innovative, and, I think, hardest solution is to find a way to make first person work with a narrative in a compelling way. I think Cyberpunk goes further toward accomplishing that than any game has in a long time, even if it fails at times (and it definitely does fail at times....I'm not at all claiming it's perfect in the way it tells its story, just that I don't think switching to 3rd person would have been the best solution to it).