The reason I asked about the crack was twofold.
One, with your unusual symptoms relating to pressure readings you have to look for unusual causes therefore I was going on the theory that there is air getting into the system from SOMEWHERE on the suction side that somehow gets closed off when the pump basket gets clogged with leaves. A hairline crack in the pump body, a leaky lid gasket or drain plug, or something to do with the plumbing between the pump and the pool are some possibilities I can think of where air could be getting in. There may be other obvious ones I'm overlooking as I'm no expert on anything to do with pools much less pumping systems. I've always been pretty good at problem solving though and your case definitely presents a challenge.
Secondly, I'm wondering if the crack in that fitting got there after the lines were capped or if it was there previously. You mentioned the pool company had told her it would cost thousands to repair the "problem" before it was decided to just cap that line and you figured you could bust up the concrete at the skimmer and fix it for much less. I'm wondering if the crack was the only problem from the start and there may not even be a problem at the skimmer. Have you considered that possibility?
Not fully knowing the history of this situation here's one possible off-the-wall explanation for your problems that you might want to explore further. The crack in the fitting on the line thats capped is
somehow allowing air into the system even with the valve supposedly having it isolated. Its also possible the cracked fitting was the original reason the line was capped instead of a problem further upstream. Unless I could see an obvious problem with that line at the skimmer end of it I think I would replace the cracked fitting, reconnect the line, fire it up, and see what happens. You could kill two birds with one stone, or, create a worse problem. In the latter case though, as long as the valve is still in the system you could just close that line off again and be none the worse for wear as that fitting would have to be replaced eventually anyway if the skimmer problem is ever repaired.
As for your priming question, yes you should add water before you replace the pump cover and fire it up. The pump is self-priming but you should attempt to get as much water in the lines as you can, although you'll probably find that it runs out just about as fast as you can pour it in. It may not be the "best" way, and perhaps an expert will chime in here, but here's what I do... catch some water in a bucket, put the water hose in the pump and let it run for a bit, pour the water in from the bucket until it starts to overflow, then quickly close off the valves on the suction lines, top the water off in the pump and tighten down the cover, open the valves, and fire it up.
Other things I wanted to discuss will have to wait until later as this post has already became a novella.
Wishing you a swift recovery.