Old lead-acid battery inrush charging current
To answer my original question, yes it is possible for an old battery to sink large currents. I believe it is caused by plates disintegrating and pieces falling and accumulating on the bottom, eventually creating a short. But this condtion usually persists past the first few minutes of charging, so it can not be called inrush current.
Now that I have thought about it, I have come across a few batteries that drew high current from a charger and never got charged up. I have even fixed a couple of chargers that failed trying to charge such a battery.
But this was not what I had. My problem was indeed corrected by tightening the belt, as most of you have suggested. Although I don't much like how tight it is now.
I did measure the charging current (thanks to mkeith who suggested using a clamp on DC ammeter) to rule out a bad battery, and it was not excessive. In fact, it was under 10 amp.
What really baffled me was that the engine ran really rough while the squealing was happening, almost stalling at times. I thought it was because the alternator load was too large. Which would also mean that the belt cannot be too loose if it can put such a large load on the engine.
However now I believe what was happening instead, a slipping alternator belt caused unstable system power - too much voltage fluctuation. This led to unstable sensor readings to the computer. So the computer was trying to maintain the idle RPM using bad input.
An alternate theory is that the squealing alternator belt presented a rapidly changing load to an idling engine, and the computer struggled to keep up with the changes, resulting in under/over corrections.
Once I tightened the belt, the squealing and the rough performance at start up both went away.