Yes, most of the MCUs today have built-in counter-timers. The problem I see is keeping them synchronized. With a single clock, you don't have that issue. Also, if you start the clock on the fist detection, you know which detector fired first. With 3 or 4 free-running clocks, you'd have a nightmare synching them and determining which started first.Choose the right processor, and these count-capture units will be built in. They'd be doing snapshots of a free-running 16 bit counter, and none of them would start it from zero. But it would be trivial to turn the 3 or 4 captured quantities into differences from whichever one triggered first. Having a processor and then building external hardware is like "buying a dog and then barking yourself".
I know building a clock out of logic chips seems a little archaic, but sometimes old-school is the best way. My main schooling is in physics and I spent a lot of time in the lab. Because of this I look for solutions that try to eliminate sources of error and that reduce procedural complexity, even if they appear to be a lot more initial work. Experience has taught me, the hard way, that it is far easier in the long run to reduce error and complexity in the design than try to fix it in the data later.
I'd put the analogy more like 'Having a processor and then building external hardware is like having a car and buying a trailer to haul a big load rather than try to stuff it all in the car.'