Since ASIs and statistics have been brought up (though not in relation to each other), I should mention that for me, I prefer to have some ASIs come from race as I see the ability score distributions of each race as normal distributions with similar standard deviations (for simplicity) and different means. I suppose I think of it in this way as the original 3d6 ability score generation method also approximates a normal distribution, and since 3d6 is used for each race, the standard deviations are also the same. Racial ASIs then represent a shift in the mean, so the smartest gnome is always smarter than the smartest halfling.
(Aside: 4d6-1 was to me a way of randomly generating an exceptional individual within the overall population distribution, which remained unchanged.)
I guess the move to standard array or point buy as an ability score generation method breaks this causal link (even though the standard array and to a lesser extent point buy is based on the 4d6-1 distribution). The focus is now on the areas where the various distributions overlap (and they do overlap quite significantly) as the range of possibilities for PCs. If you are an Int 15 halfling, you are considered an exceptionally intelligent halfling, and if you get another +2 Int from a floating or background ASI, you are among the most brilliant of your race. If you are an Int 15 gnome, you are considered above average in intelligence for a gnome, and if you get another +2 Int, you are considered exceptional.
This makes ability score generation more outcome based than a process sim, and while it is not my preference, I accept that it is the official rule since it is what many players want and avoids sensitive race issues.