I'll post this Tom H post from another thread. It contains some important info on this discussion, as well.
https://forum.sunbeamalpine.org/index.php?threads/cranking-amps.29896/#post-216397
I'll repeat what others have said:
The red light next to the ignition key should go OFF when the battery is charging. I do not know of any way that it could be mis-wired to be ON when charging. Your mechanic is wrong! I think the owners manual may confirm this, if you do not believe the people on this forum.
At idle speeds, a Generator doe not put out enough voltage to charge the battery. If you rev the engine up to about 2000 RPM and the light stays on, you have a charging system problem. An alternator can put out some charging current even at idle, but not a generator
You say you have "checked the grounds". Not sure exactly what you have checked
but understand that the ground between battery and starter must be a HEAVY cable to handle heavy starting current. At the battery there should be a heavy cable that grounds the battery to the chassis. AND at the transmission rear bracket there should be a short ,heavy, braided strap that connects the transmission box to the chassis. This cable is not described or shown anywhere in the WSM and might have been left off by previous owner or a mechanic.
If you have a voltmeter you can test your starter circuit. Put the black lead of your meter on some very good ground on the engine. not the chassis, put the red lead on the solenoid post that connects to the starter. turn the ignition switch to the start position and see how many volts are showing up on the meter . That's how many volts your starter is getting (unless the cable from solenoid to starter is bad , or the connections are bad).
Your problem may also be corrosion and bad connection between the battery posts and the battery cables. After trying to crank for a minute or so, feel the posts on the battery. If either one is warm, there's you problem. Do this test with no charger or booster cables connected to the battery clamps.
Tom