The problem is the bell curve's low granularity in the most common area will engender the opposite of what I want.
With bounded accuracy, we're supposedly looking at a "regular character" having somewhere near a 50/50 chance to succeed at what they are doing. Being a little bit better (say a +1 weapon) gives you a small increase - 5%.
Now let's look at the same thing with a 3d6 system. Rolling an 11 or higher is 50% both with d20 and 3d6, but if you add a +1, it's a 55% chance with d20 but a 62.5% chance with 3d6. The change for common rolls was 12.5%, two and half times it was for a d20. A -1 does the same thing, dumping the chance to 37.5%.
This loss of granularity makes every math advantage extremely important. Any inequality in character builds or optimization will lead to a wide discrepancy in what characters can do. You lose out on the character who only has a 16 ability score when others have 18s because it's crippling. Everyone will always be going for the races that best match their class. Trying a skill that isn't aligned to the ability scores you are increasing is so much more likely to fail that it's likely not worth taking, and the flip side is that expertise is virtually guaranteed because a +3 is godlike.
It's really the exact opposite of what I want, where players can make the characters they want and not have to sweat if they are a point or two behind or ahead of other characters.