So I may have a winner here. Behind the wall cover was a Diplexer that was taking the only coax and splitting it to the SAT and CATV/ANT jacks. Pretty safe bet that this is my issue. I'll test it when I get somewhere where I have line of sight to a satellite. Probably a few weeks out, but I have renewed confidence.
So, to answer one question, the continuity tester was able to send a signal through the Diplexer, so while that is useful for identifying cables, it doesn't seem like a complete troubleshooting option.
On your other point, The Winegard Trav'ler
manual shows two options for setting up with the power inserter.
Option 2 is what you're describing, with the PI going into both post #1 and the DVR, but option 1 (which is how mine is wired) has the PI taking the signal from the dish, then outputting to the 4-way switch INPUT post. Then the DVR comes off of post 1.
Is there any reason that option #1 wouldn't be equally good? In theory I can then hook up my Genie and 3 Genie Mini's, which I actually do have. My rig has 4 (!) TV's, so this is a potential upside, though honestly I don't expect to ever use the outside one and my kid will almost certainly not watch the one in his bonus room. (Other than for a gaming monitor.)
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2018 3950BR - MORryde X-Factor, Disc Brakes, 3rd AC, Stacked W/D, Winegard Trav'ler (DTV), 2000w inverter, EMS-HW50C
2022 Ford F-450 Limited Diesel Dually - B&W 25k Hitch