My understanding is that anything newer than a V2 rocket is just too fast for a jet pilot to actively engage.
Not quite. The part you’re referring to with V2 is ballistic missiles, specifically in their terminal (falling on target) stage. It is very hard and expensive, but not impossible, to counter these. However, only a fraction of what russia is shooting at Ukraine is ballistic missiles. Most are cruise missiles (think mini jet airplanes with wings and jet engines that fly at jet airplane speeds). These can be shot down by other jets or land based missiles pretty easily if there are defenses in place. Lastly is the Shehed drones (think mini civilian propeller plane). These are very slow flying and be shot down by land based guns, other propeller planes with guns, or even helicopters with guns. Russia is also shooting a small handful of hypersonic missiles. These are crazy fast flying from beginning to end of their flight. They’re really expensive and russia doesn’t have many of these.
That is why the “meta” is countermeasures and computerized systems (e.g. patriot missiles) where the human involvement is to approve launch for liability/Geneva Convention purposes.
I think human operators are more to prevent shooting down non-combatant aircraft like commercial airline and civilian planes neither of which are broadcasting Friend or Foe signals to air defense operators.