Engine will not stay running

Hello, it sounds like the front catalytic converter has melted down and is blocking off the exhaust. If the exhaust is glowing red, that's a bad sign the converter has failed. They usually fail due to misfiring cylinders where fuel is going down the exhaust and causes the converter to overheat and melt the substrate. Also, a faulty oxygen sensor that is not responding correctly can cause a fault in the fuel system, over fueling the engine and causing the catalytic converter to melt down.
One way to test this is to remove the front oxygen sensor and do a back pressure test. Anything over 1.5psi is not good. Even with removing the front oxygen sensor should make it start easier because there is a place for the pressure to go. If it's that bad where the exhaust is glowing red, it's dangerous to drive. They can get to extreme temperatures and catch fire.
Did the mechanic let the vehicle go in that condition?

https://www.2carpros.com/articles/bad-catalytic-converter-symptoms

https://www.2carpros.com/articles/how-to-test-a-catalytic-converter

Yeah, there's a reason why the Catalytic Converter went bad. Also, if he takes out the converter it may set your check engine light on permanently. I'm not sure what the inspection laws are in your state, but in most if your check engine light is on you won't pass inspection. I understand it may save money, but it's against the law. So, you may want to look up what the emissions standards are for your state before he does that. Just a heads up. A converter will go bad if the air/fuel mixture is too rich with fuel. Your car has a front oxygen sensor and a rear oxygen sensor. The engine computer (ECM) uses the rear oxygen sensor signal and compares it with the front oxygen sensors signal to determine the catalytic converters efficiency. Most likely the ECM will set a code P0420 and turn on the check engine light.

https://www.2carpros.com/articles/how-emission-control-systems-work