This is very odd. In the pics, I see your pump, check valve, two ball valves (skimmer and MD) and that should be it. You've acknowledged that water can indeed be seen flowing via garden hose from the pump basket to the poolside skimmer (or the other way around). So there doesn't appear to be any blockage on the skimmer line. I can only see two options from here:
1 - Try to do a pressure test on the skimmer line (just that one for now) to see if it holds pressure to rule-out a leak
2 - Something about your pump simply is not pulling water (impeller, diffuser, something).
I'm not aware of any specific troubleshooting to try on the pump at this point except perhaps fill the pump basket with water again to include as much of the suction line as possible. The check valve, if operating properly should prevent water from dropping. Install the basket lit and turn-in the pump with a full basket of water. It should get sucked into the pump right away. If not, then your pump would appear to be the culprit, but we could call-in some other pool hydraulic experts to validate.
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And to be sure, you have inspected (with power off) the pump impeller area to ensure nothing has it wrapped-up, tangled, etc) right? You already confirmed the multiport valve on your filter in in the filter position. Maybe try changing to backwash or rinse if the other things don't work?