Don't see the issue here even with the ridiculous Sat 3pm blackout (which should be scrapped, but won't). They just need to use some common sense for the later KOs and also bring them forward slightly.
Fri night 7:30pm - ensure it is between local rivals (a London derby, or even something regional like Everton v United, Burnley v City etc). And of course neither team can have played in Europe that week. I'm pretty sure they could fill most weeks with this fixture time even if they "use it to get rid" of obligations to show certain teams a number of times.
Sat 12:30pm - horrible fixture time but its here to stay
Sat 5:30pm - best time for a game in the whole week
Sun 12:30pm - if it works on a Sat it can work on a Sun
Sun 3pm - no blackout on a Sunday
Sun 5:30pm - just move the 430 back an hour, but I don't like the idea of a game after this on a Sunday to be honest
Mon 7:30pm - many weeks this is not possible of course due to Europe, Internationals, League Cup, but I am sure they can get at least one of the Friday or Monday games scheduled every week.
That's 7 out of the 10 games a week televised best case, realistically the Fri and Mon games won't both work on the same weekend but still easy to get 6 games a week with 5 at an absolute minimum. Surely thats enough and I don't see what the big drama is. I guess they could play a game on Sat or Sun night at 8pm for those cases where neither Friday or Monday doesn't work, but I hope not.