My Experience with one cable
that with OEM equipment and one cable and a simple splitter at dish running cables " only to " Vertical (or alternatively Horizontal)
Polarity outputs on LNB for F1 and F2 satelites and then splitter behind receiver with cables to both imputs one would be able to watch or
record, the channels on that polarity on all satelites.
(If you cannot run extra cables, instead of "zero" tv you would at least have a pretty good selection)
Just for info purposes, on this theory.
I run a version of one cable setup, but not with a OEM setup dish and LNB.
Mine is a 10ft old style movable mesh dish (37 yrs old I believe) with both single port, C & Ku LNB's.
While I do have two cables, one Cband and one KuBand the Cband one connects to a Cband receiver that allows
changing of Polarity manually.
The KuBand cable connects to LNB at dish and then cable behind receiver connects to multi port splitter
( 1 in and 6 out w/power pass on one port) with 4 cables out connected to both a 530 and 630 receiver
(I have ran a 2nd 630 as well for total of 3 receivers)
With this setup one can watch for example all Vertical Polarity channels on F2 satelite and record 3 other shows
as well all at once. You just cannot watch or record Vertical and Horizontal Polarity at the same time,
you just get a lost signal notice on screen.