What is the best to confirm that the output of a matched filter for a linear FM chirp is correct?

If all you want to do is simply check that the output of the matched filter is correct, do it without noise. The autocorrelation should have a distinct shape that you can look up anywhere online. It is sinc-like and will have a general shape like this

The pulse width and the bandwidth of the chirp will change where the nulls are located and the width of the lobes, but the general shape will remain the same.

In the absence of noise, the output of the matched filter is exactly the autocorrelation function of the input signal, delayed in time so that the peak autocorrelation is at the chosen sampling time (cf. the first part of this answer on this forum). I don't recall off the top of my head what the autocorrelation function of a linear FM chirp signal is, but the sharp peak that in your bottom trace suggests that the autocorrelation function is sharply peaked, perhaps like a sinc function with very small null-to-null distance.