When I had c-band, there were a lot more satellite locations. The receiver simply moved the dish to find the correct satellite for the channel you requested. Your switch technology then allowed that signal to go to multiple receivers. Echostar was the company who was a leader in building that type of receiver. The technology exists to move the dish. RVers do it all the time. The little E* dish should be a lot easier to move than the 3-4 meter c-band dishes. The major problem you have is that all the receivers will be limited to the channels on the selected satellite( 24 c-band and 36 ku-band). Some how millions of c-band users survived with this technology. I remember if you were watching a station at the east limit and then moved the dish to get a station on the west limit , it could take a minute or two.
If you use a movable dish 1000+ , you could see 4 satellites at the same time. You would just need to set the satellite spacing to be the same so on each move the 4 lnbs line up. This way hundreds of channels are availabe on each move. However, not all may be available.
Many OTA antennas also use a rotor to move the antenna to face different channel's tower locations. It has been a long time since I have used one. I remember the rotor control had to be set by hand to move the antenna. If you had more than one tv connected to the OTA antenna you were also limited in reception quality to the stations the antenna was facing.