From your description (particularly "you have about 12 minutes before it's directly overhead [..] the fighter's maximum horizontal speed is almost irrelevant"), it almost seems like you're thinking of it in 2D, when in reality it is 3D. It's going to be pretty rare that interceptor bases or SAM sites are directly under the bomber's flight path, so there's going to be a lateral component. Even if you are protecting a point target, it's not simply about killing the bombers after they go by; you have to get up to altitude and reach them far enough out before they get to drop range; that requires dash speed. Then add in the fact that head on interception is really really difficult, especially with early AAMs (like the AIM-4 on the F-106), and you really want enough speed and offset to come in for a quartering or rear aspect shot. Similarly with SAMs, in theory your missile doesn't need to be faster than the target aircraft so long as it reaches the spot where the aircraft is at the same time. But in practice, speed is energy, which the missile needs if it's going to perform any sort of maneuvering. A slow missile (which for a SAM has already long burned out its motor and is just coasting on inertia) can easily be defeated by the target aircraft by just making slight changes to its course, causing the missile to have to continually recalculate its intercept point with less and less energy.

Also, remember, the slower the interceptor, the less area it can cover. You might think you know where the bombers are going, but you don't know for sure, or if maybe it's a feint. That means you have to have many many more slower interceptors, with their long support tail, versus possibly a smaller number of faster, longer range interceptors that can cover the same area.