Rental contracts deal with property, which in many states requires a writing to form an enforceable contract. As you say, property contracts are controlled through the Statute of Frauds adopted in most states. Most other agreements are not.

If I say to Alan "scratch my back and I'll pay you a dollar" and Alan agrees, we have formed a legal, valid, enforceable contract. Once he scratches my back, I owe him a dollar.

Unwritten contracts are just much harder to prove than signed written contracts. But if we go before a judge and I admit that that was the agreement but that I changed my mind after I got my back scratched, I'm going to lose.

Yes you could I suppose...my point is a second look may prevent such occurrence.....or not!! :)

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I believe you ask the wrong q.
I believe you could benefit by re-framing the scene...
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A contrarian view:
Our generator is on the roof.
The fuel is sunlight.
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Nearly two decades full-time live-aboard, we never saw much need for a petroleum-based genset... busted or otherwise.
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Facing your situation, I would:
a -- invest in batteries before I
b -- invest a nickel's worth of energy fiddling with the busted (aka 'toes-up') factory-installed genset.
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The seller is out of the equation.
Become friends with exploring My New Reality.

You might want to check with an attorney, I check with mine all the time with tenants "trying" to pull a verbal contract. Contact law applies to contracts being tangible (real) or intangible. From first hand experience that generator unwritten purchase "agreement" will never see a day in court. The buyer accepted delivery and tried to make claims weeks later. Ask your attorney, not the internet.

I'm a retired lawyer.