I suggest monitoring closely the first time. A 2000 volt supply, while experimenting, went up to over 3000 volts no load. Capacitor breakdown failure happens quickly.That makes a ton of sense! It will eventually blow past the SCR which is the only real problem with it other than safety. I will put a 1 KV rated SCR on the device that should honestly fix the issues. I am also going to add a bleeder resistor. It takes several seconds for the capacitor (4.7uf) to saturate with 800 volts.
Leakage inductance and poor coupling between windings are the two sides of the same coin! That may be the motivation for splitting the secondary into multiple sections. (see post #7) and also for adding quite a bit more material to the core, so that more of the magnetic flux passes through all of the secondary. We can be sure that there is a benefit if somebody who knows goes to the extra effort and expense to make a better transformer.An inductor on the output between the rectifier and the capacitor should deal with them (provided that it doesn't have too much self-capacitance, which is often the case with high-voltage high-inductance components with lots of turns of fine wire).
I'd agree with @MisterBill2 on the Motorola design.
The spikes are flyback pulses cause by poor coupling in the transformer. I presume you have two primary windings. Are they wound bifilar, or concentric, or one either side of the secondary?
A H-bridge circuit might work better.