scag stc48a stalling/starting issue

Father-in-laws machine, he said it would not start, but when I got on it it started fine. Then he said it stalled out when mowing, again I restarted it fine. He said the battery is very old, but I am not sure how that would make it stall out when mowing. I am wondering if the alternator on the engine is not working, is there a way to check that? If it is not, is it possible that the battery would be able to run the engine until it was drained? I am not familiar with these small engine charging systems. I know there are 3 interlock switches to allow it to start, I had to replace one last year, but I believe these only are used for starting, and once running have no effect. Also, there is a seat switch, I assume if that is flaky that it could cause the machine to stall, but is it involved also with restarting it? Sorry for the rambling.. I guess I have these questions:

- can a bad battery make a running machine stall?
- is the seat switch involved with starting?
- is there a way to test the alternator? just a meter on the battery + terminal?

yes
no
yes

There is a fuel solenoid on the carb that stops fuel entering the main jet when you turn the engine off.
If the voltage drops below 9V - 11V ( depends upon model ) the solenoid closes so the engine stops.

Simple test is voltage across battery terminals
1) everything off
2) ignition on
3) engine running full speed
4) engine running full speed blades on

Thank you. I was finally able to inspect it, the battery appears to be 10 years old, so I will replace that to start with. That seems way past it's expected life...

Test the rectifier output for AC Voltage first.
If the rectifier is bad & is pumping AC into the battery it will damage a nice new battery very quickly.