car stalling out: more than just the battery?

[BerniniCaCO3]

Hello!

I own an old ford myself, but I'm now the "car guy" to my friends, and one of them has a 1989 740 (dl? some trim-level suffix) with about 130k on the dash (which does still turn over, so likely accurate).
This friend is driving it into the ground, and keeps it at a nominally operational level. It still has a split front swaybar that he's not fixing, because it still gets him 3mi to work and back. But that's neither here nor there.

It wasn't starting for him.
It seemed to be cranking slowly, and in fact, we were able to start it about half the time when I went to look at it. We later jumped it, and it cranks a little faster/ started every time.
It starts,
but, it stalls out almost immediately unless you keep on the gas, even if you jump it. If you keep it at 1000, 1500rpm it's just fine. Let it drop to an idle even after 2-3min on the gas, and it dies again. Or if you put it in gear.

Obviously, the battery was weak. He says it's 1-2yrs old. It could start the car, but barely-- so it probably wasn't DEAD dead (probably did not have a shorted cell). I didn't have a voltmeter handy.

I noted, when I let my friend hold the gas, that the belt running the alternator was flapping loosely. Aha! right? Not charging! The a/c compressor had locked up, and so someone had purchased a smaller belt and routed it just to the alternator bypassing the compressor. Luckily the alternator has its own tensioner (just a threaded rod that you advance to pull the alternator out).

So I tightened it to a good tension. It no longer flaps at all to 2krpm, but, at 3krpm you see a 1/2" of vibration. Maybe it could be tighter, but it should be tight enough to charge.
It still stalls out.
So what I don't get, is that what started out as a cut and dry charging/starting concern, is maybe complicated now.
If it were the alternator belt, then tightening it should have worked.
If it were the alternator itself (regulator, usually, to be specific), then keeping rpms higher shouldn't have helped. Unless a regulator can go out in such a fashion that it only produces sufficient voltage at higher rpms? A voltmeter might be helpful here.
And if it were the battery being so drained, that it's pulling down the voltage on everything, even when the car is running, then it shouldn't have been able to start the car at all, and it could-- just with a longer crank.

I'm worried it might be more.
There aren't any engine noises that stand out and scream to me that something major is wrong with the engine, and once it's up and running, seems smooth/ no missfires or hesitation.