I just recapped a IIcx (close enough to a IIci) that would not power on and it works fine now. Mostly I just replaced the (9) 47uf 16V analytical aluminum dome surface mounts since I didn't have 10uf ones for the last 2. IIci's have the same issue as do other macs of that vintage with the power on circuitly apple used (never had a issue with old PCs that used a generic on/off switch and are older).

If you want to use a non mac monitor with the old mac II series with built in video you should dig up any Sony CRT monitor you have since they do sync on green and also do all the Apple video modes that most PC monitors don't bother with.