You keep saying the heater control valve doesn't open with vacuum, but unless I'm misunderstanding what Ron D said, it should be open if there is no vacuum. Vacuum closes it. So it seems to me like if you had no vacuum it would be always open (unless it's a p.o.s.) To repeat, vacuum closes it only if selector is on Max AC or Off. Surely someone on here knows and can confirm/deny that.
As to no pressure, I'd check/replace the radiator cap. That's where the pressure relief valve is. If it's failed, it could let coolant go to the overflow tank at low pressure thus causing what you report. But you'd still get heat if the coolant is hot, so, two different problems. I doubt replacing radiator cap would have any effect on the heat problem. But it's cheap ($10 +/-) and 5 seconds to change so I'd do that first.
Just to be clear, I'm no expert on this... just following what seems logical. The h.c.v. has to be spring-loaded to return it to open at no vacuum, so if yours sticks closed, it sounds like it's trash. It would make sense that the default would be open, because you can live with the core active even when using AC, but if the default (no vacuum) was closed, and it failed, then you have no heat, and can't pass inspection, if you have that, not to mention it's a nightmare in winter. Mine, with engine off (no vacuum), if I push a bit on it, it springs right back to, I assume, open. So, loss of vacuum, if it were that problem, would cause it to be always open. Sounds to me like yours has a broken/failed spring and if it gets closed, it stays closed... guessing... mine's different because as I said before it's only on one hose, but in operation they should be similar. What I'm saying is, if you can move it back and forth by hand and it doesn't return to its home position (open) then that's not how it's supposed to work so it's crap.
I suppose you could take the heater control valve right out of there and (with connectors) just connect the hoses, that would tell you for sure if it's that or not. The only downside to that is you'd have hot coolant going thru the core on Max AC and Off, but I don't see that would cause any particular problem except at Off you'd have perhaps some heat build-up around the core... probably wouldn't matter. I would imagine the reason it works the way it does is that even when AC is on (not Max AC), you still want the heater core active because sometimes you just want mild cooling.
I'm sure some big guns on here will chime in as to whether what I'm telling you is good info.