1. Most tasks don't really boil down to a single Stat. Think, for instance, of climbing a sheer wall. Is that DEX because it requires agility, STR because climbing requires you to support your own body weight, CON because it requires you to be fit, or INT or WIS because it requires you to think about where handholds are and apply climbing know-how? Roll-under-a-Stat, then, seems a little reductive.
2. It puts emphasis on thinking about Stats, which isn't as bad as obsessing over skills, but still can be restrictive ("I won't bother attempting difficult task Y because my character's Stat X is crap").
3. I think I would find myself using it as a lazy shorthand to stop me thinking carefully about PC actions in the game: "Oh, you want to negotiate with the guards? Roll CHA."
I prefer nowadays a method which I think is more intuitive and more humanist (!). That method is as follows:
- Player says their PC might do something
- DM considers the situation in general (including the Stats of the PC) and assigns a target number for success between 1 and 6
- DM tells the PC what this number is
- PC decides whether to go ahead or not and has a small amount of negotiation