I posted a picture of my single-hole skimmer a while back trying to find out how I could turn it to allow suction from the main drain, but I don't think anybody had seen such a skimmer before. I believe it is an old style "Anthony" type skimmer as illustrated here (picture of the part that I believe is broken). The tab that allows you to turn the adjustment is long broken off, leaving no way to turn it. This means that backwashing is impossible except for in short bursts until the water level goes below the skimmer and I have to wait for the hose to refill. I also have no way to drain the pool in any meaningful way except to use a sump pump that would take a day or two just to get 25% of the water out, and I'd be unable to run the filter for that time.
In short, I have to figure out how to "fix" this or work around it before I have to tear up the concrete and replace the skimmer (which means I'll lose my budget for tile when I resurface).