Any suggestions on how I'd isolate and test the pump motor and pump?
I knew it was going to get deep and complicated! On reflection first thing I'd do is confirm the pump motor is getting power when it should be with a volt meter. In other words when lever lifted does the pump motor get power. Or if you disconnect the auto-fill level sesnor (like for boiler over-fill when descaling) does the motor get power. Fairly certain motor should be getting 110v AC. If getting power and the motor does not run likely bad motor, possibly starter capacitor. IIRC some motors the capacitor is internal, some external. I don't know how to test if the capacitor is good or bad off the top of my head, don't work with AC motors for a living and a long time since college and AC classes. But if the pump motor not getting power when it should I'd rig temporary AC to motor and confirm motor good. But the bad would be if and why the motor isn't getting power when it should versus motor replacement which would be relatively simple. Since it seems not running from switch activated by lever or when "told" to fill the boiler it's getting deep into troubleshooting. Don't know near enough to attempt to teach how to figure it out via the Net. Don't know if all commands to the motor go thru the Gicar brain or if lever switch direct to motor without Gicar control also does the auto-fill circuit go through the Gicar. Might be obvious tracing the wiring might not.
Of course the pump motor could be running and the pump not pumping...'nother area of troubleshooting I'm not qualified to attempt. I have some ideas what I'd do but not practical hands on experience.
Now why the heater isn't being "told" to come on (no relay click) I don't know. Again could be low boiler water level and solved when pump back running or could be it's brain dead Gicar. Heck or could be shot pstat not triggering the relay but since pump has issue pstat wouldn't be my first probable choice for no heat.
Sorry I can't be more help. I'm a DIYer not a professional espresso machine tech working on these things everyday. If you don't have the expertise may be time to pick up the phone...