How you describe the relative wind angle doesn't really matter. Personally, if I call close-hauled 45 deg relative, I'd call a dead run 180 too. I wouldn't get too exercised about it.
Okay, the usefulness of a genny on a broad, broad reach or a run (say 150 to 180 relative. It's nearly useless at 160, almost totally blanketed. At 180, it's useful again, now you can wing it out. A spin. pole will help keep it there, so you can still daydream (a little) while steering. But if you want to go fast straight downwind, bite the bullet and get an old-school symmetrical kite.
So let's say your rhumb line course is 150, and no kite on board. What to do?
Just what you suspected...head up to 130-140, where the wind's still coming "across" the boat and your genny will fill, then alternate heading off to 180, where you can sail wing-on-wing. Repeat, then repeat again, as searoom and depth permit.
If you don't like this answer, then get a spinnaker, and steer 160 to the wind all day.