So despite the airy nature of the preceding, I do have some concrete thoughts on why these things matter. Neverthless, this will manage to ramble. You've been warned!
Critical hits in D&D provide a good example.
To begin with, from the "proper abstraction" perspective, critical hits do not belong in most D&D rules, in any form. Strictly speaking, the attack roll is a scaled check with a binary outcome--you bypass the Armor, dodging capabilities, magic, etc. of the target, or you don't. If you so "hit" then you roll for (meaningful) damage. If you roll high for damage,
that would be a critical hit.
Note that I am here talking about incoherent abstraction, and this applies equally whether the rules for attacking and damage are process-driven, result-driven, etc.
Perhaps it can be seen most readily from the process-driven perspective, though. Roughly, you roll your attack to see if your swing connects, and then you roll your damage to see how well (based on the weapon, your STR, luck, etc.) A critical defined as "double damage on a 20" or "max damage on a 20" or any other such variant is suddenly violating the fundamental abstraction to make how well you connect matter on a 20. Thus the "process" becomes "how well you hit usually doesn't matter, but occasionally it does," and this is perfectly capable of producing results that people find a bit counter-intuitive. You crit with a 2d6 attack and roll snake eyes, then next round barely hit with a normal attack for 10 damage, followed by an almost crit for 7 damage.
Most people accept this, because they like having critical hits, want critical hits tied in the fiction to thing closest to the swing (the attack roll), and are thus willing to overlook the hole in the abstraction to get what they want. Max damage on a crit is one of those bits of tap dancing to try to finesse the issue.
If you stop here, you can probably get away with it. But the pressure to not stop here is immense. Already we hear it with Next. "Max damage on a crit is boring." Is max damage on a non-crit boring? Then why is getting an extra dose of it without even needing to roll the damage dice suddenly boring? Abiltiies to crit more often, up the damage, etc. will all work mechanically, but every last one of them will expose the hole in the abstraction. In fact, some of them will
only exist to try to deal with the hole, the prime exhibit being the 3E confirmation roll.
Don't find that compelling? Let's try this thought experiment (not a proposal). Go back to early D&D, when all weapons did 1d6 damage as a base. Then apply the modern understanding that it is generally satisfying to have some success about 2/3 rds of the time. OK, for all weapon attacks, don't use a d20. Roll your d6. On a 1 or 2, you miss. On a 3-5 you do that damage. On a 6, you do that damage, and this is considered a "critical blow."
(Presumably, you can apply your STR mod, magic plusses, etc. after the resolution above, to damage only.)
This is a cleaner abstraction, given the stated design goal, but the process-simulation has been cut out. You swing a sword in the fiction. In the mechanics, you roll damage, ignoring low damage, narrating it as a miss. (I'm aware it is less clear-cut than that, but work with me.)
I can state with reasonable confidence that most D&D players would not find this cleaner abstraction pleasing. So roughly I think we've established both the necessity and some of the pitfalls of the process-driven attack roll.