Chrome developer tools – Network – how to filter only POST and PATCH requests?

The filter option is quite powerful, but it’s limited to only using AND (conjunction). So simple workaround would be to use negatives…
For instance, in your case you could create a filter like this:

-method:GET -method:OPTIONS -method:PUT

This should filter out most of the requests. If you have other offending HTTP verbs you can easily add them.