Will my 1950's 240v 6-30 outlet be safe to use?

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On the other hand, my feedback (not advice).
I would ensure
1) the 10-30 receptacle is wired to code and
2) the receptacle pins have solid grip on the Tesla mobile charging cable 10-30 adaptor

My 1920's outbuilding garage had legacy wiring that I upgraded where prudent, including receptacles to modern code.
I implemented a 10-30 receptacle (bought new) because it was important for my garage panel to stick to 24A continuous current draw.

Been charging our 2013 Model S on 10-30 via mobile connector for 100k miles of driving. We avoid unplug/replug, ie, keep the mobile connector permanently plugged into the receptacle. Only in a few rare off-grid camping trips have I brought the mobile connector with us.

The importance of checking the connection of the adaptor pins to the receptacle is key, you should not allow this connection to overheat, the adaptor will warn you of

Yep I had already bought a Hubble Industrial grade nema 10-30 to be safe.

I talked with an electrician friend and his main concern was the age of the wire. The house had been built in the 1950's. We don't know (I still need to check) if the wire runs through conduit, in the walls or under the house. I have only been in the house for 5 years but apparently there was a problem with rats before so a rat could have chewed on the wires. He suggested running new cable. If we do that I am going to just run cable/ground for the wall charger, and change the breaker to a 60 amp (if the box is able to handle a 60 amp breaker). That way everything should be perfectly safe and can charge at the fastest rate.

Even if you can’t run a 60 Amp circuit run the wire for it any way and put what ever breaker you can safely fit. You can upgrade later if you ever upgrade your panel. The cost difference won’t be that much.