Can exhaust backpressure cause a car to cut out?

Car cuts out sometimes immediately sometimes after 30 seconds etc.

If I hold the key on the starter position engine keeps running and stays on.

I know this can be caused by a faulty ignition switch.

However I’m suspecting/asking if this same behaviour can be caused by a damaged resonator as I have read In one article it can. I have been experience reduced performance and sound for a while since I bought my low car from a garage. Recently I hit the resonator on a speed bump and the sound and performance improved immediately. Days later I hit it again and the sound and performance went bad again. This suggests the resonator is damaged and is causing the sound and performance issues I’ve had ever since I bought it back from that garage. Obviously they hit it and damaged the resonator while in their posession.

I had recently changed a faulty ignition switch with another used one so could be replacement switch is at fault however it when I hold it to keep car running it feels very different from first ignition switch. First one also used to start and run more often.

In any case could the current symptoms be due to a damaged resonator or is it more likely the ignition switch. How should I approach this, replace resonator first or would it be better to open and check if the ignition switch is corroded?

You can already see why I think the resonator is damaged but is there another way to diagnose that it is bad?

Thanks