In the last few years I've found that I need to pause between run and start for a second to prime the line whereas I didn't have to do that before. I've replaced the pump in the last year or so but that had no effect on the need to pause. Since its an easy workaround, I haven't really put much brain power into it but now you have me wondering if there's an inline check valve somewhere that I've missed noticing.
Regardless though, you should have no problem priming the line once the key is turned so the problem is still strange. Your statement that it acts flooded and runs rich really makes it sound like an injector problem. Since this is an intermittent issue that may or may not be related to temperature, its possible that your injector testing was not conclusive.
I would replace injectors. It shouldn't be too hard to find a set for cheap either here or on the facebook group. Even a junkyard would work although they would probably be a little more expensive that way.