[franken]

Just random BS, but get some jumper cables.
Disconnect the big red (or not) starter cable.
Start it, drive, see if the problem goes away.
If not,
Disconnect the alt big red wire at the alt.
Start it, drive, see if the problem goes away.

The starter starts the engine. Then it's done.
The alternator provides electricity to the system. It charges the battery and runs things til the engine is stopped. If the alt doesn't charge, the battery supplies current to the ignition.

Thus, the starter and alternator are not the problem unless they aren't behaving

Assuming a charged battery (2 days on a slow charger) and 20 minutes I'd be looking at the starter.
I'd also be putting my hand on cables, connections, parts, etc. to see which if ant are hot.