In this scenario increasing lead distance is equivalent to detecting and firing earlier (ie increasing distance between x1 and x2), which again has serious technical issues. Yes you can hit a falcon with the same bullet, but it's harder to hit. And the big difference in these scenarios is that the bullet is moving much faster than either the pigeon or the falcon, which means you can always buy more time for the falcon to get to the interception point by taking a less efficient path, whereas for the satellite interception where your interceptor is slower than the target, there is some optimal path (basically straight up from right below perigee) that sets a hard limit on how long it takes to get to any interception point.

If your interception point is say 500 km straight up from your launch point, just to get there you need a rocket that rapidly accelerates to over 3 km/s, and it takes a little over 5 minutes to get to the interception point. To hit an 8 km/s target, you need to fire when it is 2500 km away, to hit an 11 km/s target you need to fire when it is 3500 km away. If to score a hit you need to be within 10 meters of the target at interception, then you need to know the position and velocity of the 8 km/s target to within 4 ppm, you need to know it with an accuracy within less than 3 ppm. And note, if you are shooting from 15 km up in a fighter jet, the 8 km/s target is within your radar horizon, the 11 km/s target is not. Not to say that you couldn't build a system with target sharing and more precise tracking, but it would be a different system than the one you need for the less challenging use case.

Now it's not a strictly either or thing. You could use a missile with intermediate speed and fire it less early. There's a whole continuum of solutions, but it will always be harder to hit the faster moving target. And of course you have real world things to consider like if the target has any maneuverability or countermeasures and how much variation in conditions you can afford. Generally you are already using the best targeting system you can get and are already in the most advantageous position you can be in, so the knob you can turn is "how big of a missile do I need to get the performance I require?"

Agreed, it all depends on the use case. If you are trying to hit an evading target, you have to be faster. If you are trying to deny the ship from a destination, you don't.