Once you have a team, it's pretty rare to have a "quantifiable" impact in the way that satisfies your engineer side. By design, your time is pulled to the problems that can't be solved quickly/definitively/numerically, or they'd have been solved by the other members of your team before you. And to the extent that solid metrics can be cited, you want them to be your team's accomplishments. The lead who takes credit for everything is a jerk.
I'm not saying your take is wrong here (some people seem to love this kind of management porn stuff a bit too much and roll around in it), but it is true that as you get more senior, you're expected to do things that don't have that kind of quick payoff.
Lead Engineer and primary designer or implementor of a service that handles 1M queries per second with 100ms 99%ile latency, globally distributed for 100M users.
I'm not saying your take is wrong here (some people seem to love this kind of management porn stuff a bit too much and roll around in it), but it is true that as you get more senior, you're expected to do things that don't have that kind of quick payoff.