I'm sure there are groups for whom this is a good approach, but in general I don't like it. First, rolling the skill checks is fun for the players--taking that away from them will, all else being equal, make the game less fun.
Second, while there is room to use this well for fudging/handling impossible attempts without directly saying "no," I think it's more likely than not to make matters worse. It can be used to cover fudging that's contrary to the norms of the gaming group--I don't want my character to succeed or fail based on GM fiat when ostensibly making a skill check dependent on character skill. And it would make me suspicious that all of the dice rolling is a cover for GM fiat. GM fiat can be a perfectly good resolution system (see Amber DRPG for an example). But GM fiat pretending to be a DC, dice, and skill modifier system strikes me as likely to produce toxic player-GM interactions.
Third, this seems likely to slow things down, which again is likely to be a bad thing.
Fourth, I don't view figuring out the DCs as being inherently bad or contrary to immersion or whatever. That's how the game encodes difficulty. Sure, there may be a little aspect of "I want to think of it as a slick wall, with some natural crevices but without good hand and foot holes, not as a DC 25 wall." But I find the opposite problem is more common--I've gotten a description that I can picture, my character should know roughly how hard it is for her to climb the wall, but I can't use the description and my character's Athletics (or Climb, or whatever) to figure out how hard it is. In that environment, having a skill check where a player says "wow, if we keep trying that we're going to fail badly" based on calculating the DC isn't a bad thing for immersion, because the characters could likely reach the same conclusion.
So yeah... done well and in an environment with iron-clad trust, this could make for a more immersive game that might be more fun. But for most groups most of the time, I think this will be negative.