No. The lubrication of an engine does not mingle with the transmission in any way. It sounds to me like that particular vehicle was not maintained and all the failures are expected, though not related. The transmission needs some cooling for reliable operation and it is lubricated by its fluid. Transmission fluid sheds heat into a section of the radiator but those fluid do not mix there. An overheated transmission can fail from being too hot, so a failed radiator has at least that influence.

No. Having no engine oil would be a major contributor to your engine going bad.

The radiator isn't going to go bad just because of no engine oil and the torque convertor doesn't run engine oil, it runs transmission oil.

Yes- no engine oil will make a radiator and a torque converter go bad, right? Yea, that's the ticket....no oil....

The engine would overheat due to friction and the cooling system would fail just before the engine. The torque converter would not suffer any ill effects.

hmmmm.......No oil at all - engine would run Hot for a short while and then seize.....conceivably raise coolant temp a bit before it blows.....but can't see radiator failing from no oil.

Torque converter: many transmissions are cooled by the radiator. More likely, the radiator failed and caused overheating of transmission/torque converter.

Whatever - sounds like you scored a home run.