I think you are confused.
Well, that's ironic.
There is no requirement that something be the only reward in order for it to be a reward, and therefore affect behavior. And I think you will find that people in D&D tend to enjoy character success.
Yes, and...
"Character success" does not directly map to having inspiration; it's just one of many factors. And, of course, "character success" is only one factor in the larger goal of having fun. Players are not all min-maxers, and not all players always choose the action that is most likely to lead to success.
Will there be
some incentive to make more rolls, for the purpose of getting 20's? Yes. Will
some people respond to this incentive more than others, at least when first trying out the rule? Yes. But, unlike "pay for quantity", there is not one factor here.
But more importantly, there is a huge difference between externally imposed metrics/rewards, and the rewards we give ourselves. The desire to have fun, and to have character success, comes from ourselves. It is not an externally imposed metric. If making extra rolls isn't fun, people won't do it, even if it can lead in a roundabout way to another facet of fun.
I'm not denying that the mantra "that which is not measured is optional" is relevant to a work environment. I've seen firsthand how true it is. I just don't think it applies well, or maybe at all, to gaming.
Also, please see the example I keep bringing up about murderhobos, a well known phenomenon, and only XP for killing in several editions if you do not think a character-focused reward will motivate players.
What do you mean by "well-known phenomenon"? Documentation? It hasn't been my experience. Sure, there's an incentive to kill all the monsters in the dungeon, whereas milestone leveling makes it more likely that unnecessary monsters are skipped (although, even then, players think there might be some treasure). But I have never once been in a game where a player wanted to kill innocent farmers for a few extra XP. Wouldn't your theory predict that such behavior would be commonplace?