Just read over your post.
You said it runs on starting fluid. That clears out a lot of things.

You are correct about the relays not turning on the pumps it appears or it's a corroded fuses in the panel problem.

When you jump the pumps you can start the car because the relay is not running the pumps.

What I don't know is the part about how the fuel injection gets turned on?
If it doesn't then the car would not run but starting fluid makes it run only for seconds.
I know the relay has two contacts and I thought one was for the FI and the other for pumps.
Either one, not closing, means it won't run on its on!

You said the relay clicks in your hand but that does not mean that both contacts close but the relay
is working in response to the CPS, ICU or ECU.

Most of us go over the circuit board with a soldering iron and touch up (reheat) the solder lines and add some fresh solder. Make all of it, look shiny again. Thicker can help!

Another quirky thing about relays is the magnetic coil gets warm. If it should become overheated, by aging from shorting turns, it will lose its magnetic field and drop out of duty. Carry a spare! Even the repaired one can be a backup!

Be aware that the terminals connecting all this can get hot as well, so, Check the socket terminals for discoloration or corrosion.

If you ever open up the instrument cluster panel there is a Main headlight relay, in the dash behind and under the left vents, that can suffer the same fate!

I hope that the relay is the problem for you and not 200k old pumps.

Phil