If it's a thing for checking performance, like "your site is up but very slow", I have lighthouse checks as well, plus I write response time to database.
Just took a look at the site which had troubles, it has 2-3 seconds response time. But this is only for getting first host, not js/css resources etc. So probably indeed sometimes it takes more than 10 seconds to get response...
If the site is slow you can send a different alert. From that screen they can be prompted to configure settings, disable the alert, set timings, et cetera.
2-3 seconds to receive headers is excessive. Anything over 1 second needs improvement. Realistically Googlebot isn't going to crawl much if it takes more than 500ms.
1. By default I'll setup timeout to be 30 seconds, which will be customazible. Maybe 30 secs max.
2. Add ability to setup alarms for slow response time
Cool!