Where did you tap in your button(s) in relation to the ignition switch?
If you went for the junction where switch plugs into harness, then everything is still connected and should work as intended.
ie: run will allow it to run, pushing the button engages the start circut which also deals with the CSI (if the thermo time switch is functional and still cold enough to allow the CSI to spray and its the 1st try at starting and the starter hasn't been cranking for around +10 seconds) and the fuel pump bypass circut that allows the pumps to run during cranking.
IOW, if you tapped in under the dash where the factory switch plugs into the factory harness, then everything 'should' still work just fine unless you did something wrong or there is an actual failing in the cold start injectors circut or its associated sensors/devices.
Having to hold the key in the start position (or trying to arrange it so) in an effort to get the CSI to work is not the way to bypass the switch (which you say was faulty, so why even revert to it as part of the new system to begin with?) and will just give one more place for a potential failure of the circut while you're buttoning and holding and stuff to try and fire the engine.
A correct bypass of the factory ignition switch would take the switch out of the system entirely and replaces it with 2 switches: one toggle for 'run' and one hold~to~activate type for 'start'. Anything else and you're still partially relying on an already dubious switch to get the job done that it couldn't do by itself to begin with.
After switch conversion, the only thing the factory ignition switch should ever do again is lock the steering column and/or fill the hole that would otherwise look like hell if it weren't filled regardless of the lock post being functional or removed from the switch cylinder assembly.
'Course that's just my take on it and anyone can do whatever they like to thier car regardless.
EDIT:
Dammit Jim!
LOL