If the site is up and you send them an alert saying it is down, that's a bad UX.
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.
I see. This is what I use lighthouse audits for, plus I'll have different alerts which user can enable for slow sites, like "If response time > 2 seconds trigger an alarm".
So
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
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.