From your description of the fault , it does sound like the starter has an internal short .
An amperage draw test would have to be performed but that is a little involved if you don't have proper equipment to perform the test . Not much is needed but I think it's better to confirm that the electrical system is ok and replace the starter if all connections are found good .
Diagnostics has its place but if you feel the engine cranks when the starter is cool , the electrical system is most likely ok but always good to inspect . You may have a battery positive that can't carry the amperage and a ground cable that is the same . Check those first .
If they appear good , check if they are excessively hot after cranking .
I do however suggest that if you replace the starter , replacing the positive and negative cables is good to do if they are very old . Cables corrode inside the insulation and cause great resistance which can very well produce the fault your experiencing .
You may want to try that first but dollar wise it may be feasible to purchase a starter and cables and call it good . Generally , OEM electrical is often better than what we find today in parts stores so if you feel the starter has any chance at all . I still think it could be worth swapping out the cables first . Just my thoughts .