Before anything, we need to determine what has happened with your engine. Since you are questioning the head gasket, here is what I suggest. First, lets run a compression test. If the head gasket is out, you will find low compression in one or more of the cylinders.
Here is a link that explains how in general it is done:
https://www.2carpros.com/articles/how-to-test-engine-compression
Compression should not be less than 100 psi and not vary more than 25 % from cylinder to cylinder.
When you perform the test, pull the ASD (auto shut down) relay from the fuse / relay box under the hood. That will shut down both fuel and spark. See pic 1 for location.
Also, take a look through this link. You may find it helpful.
https://www.2carpros.com/articles/head-gasket-blown-test
I have to be honest, I never heard of a stuck thermostat automatically shutting the engine down. The thermostat functions based on coolant temperature and has no electrical connection to shut the engine down.
Try the compression test and let me know what you find.
Joe
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Thursday, July 1st, 2021 AT 1:30 PM