I'd replace C31, Q5, and IC23 and *hopefully* be done with it. Any/all of those 3 can cause the coils to fire upon boot-up. I had a very similar problem with my Black Knight a while back, but it was activating all the coils *and* blowing the 2.5A slo-blo coil fuse upon boot-up...but it was only doing it intermittently...and if it managed to boot without blowing the fuse, it would work perfectly fine for hours on end. Drove me nuts. Still can't remember how i came across the suggestion to replace those components for the blanking circuit, but that fixed it and has worked fine ever since. They're easy enough parts to replace, and if they don't fix it at least you know you have a good blanking circuit for the foreseeable future.

Here's the relevant portion of Clay's guide that i got the info from:

Likely Blanking Signal Problem Parts.
Remember the likely cause for a low blanking signal (other than the things mentioned above) is a timer chip failure (556 for s36, 555 for s7) at IC23, a bad C31 (1mfd 25v polarized) capacitor, or a bad Q5 transistor (2N4403). The Q5 transistor tests easily with a DMM, and replace cap C31. The timer chip is small and cheap to replace. So start with those thing. In system3 and system4, the 556 timer is also used to generate the IRQ signal, so suspect a bad C31 or Q5 first. On System 6/7, the timer chip (556 for sys6 or 555 for sys7) is only used for blanking. Also on sys6-7 look at the outputs of IC25 (4020 1ms timer) with a scope or logic probe. If the interrupt signal is not being generated at the correct speed, then 555/556 timer IC23 will not generate the required "pulse".