I can't claim to know what they legally can or can't do... nor could I say for sure that they wouldn't go beyond what they technically legally are supposed to do, by which I mean... just because they aren't supposed to be able to enter without a warrant or probable cause, doesn't mean it can't happen.
That said... Late last year I had two welfare checks on consecutive days. Apparently uncoordinated and independently two different people that I hadn't talked to in a while decided on those two consecutive days to call and have people sent out to me. In both instances I did not hear doorbells or knocking for a while due to watching TV on a different level of my home from which the main door exists.
In hindsight I wondered if I hadn't gone to the door, would they have knocked it down at some point? I can't say. In the first instance, they kept wanting to talk and called out paramedics and I had to talk to them for a while too. I mean, I don't know if I "had" to but I did. On the second instance no paramedics were called.
Randomly, on a day weeks later I ran into one of the police at a county office where I was paying a tax bill and he asked how I was doing and I told him he wouldn't really want to know.
In my case, though, I could completely honestly say in those moments that I was no danger to myself because at that time I was not. My "plan" all along has been to wait as long as I possibly could before other circumstances forced my hand... so until I get to that point, all the other thoughts are just possibilities. So they aren't going to "catch" me in a lie or anything because I have no immediate plans until such point as I do.
So, my experience might vary with others. But, I did find a certain irony to how if I wanted to talk to someone who didn't want to talk to me, they could probably report me for harassment or something... but when things are the other way around, they can apparently send police to check on me. Seems unfair. Although, on the flip side, having two cases of people sending out police and the police deeming I was fine might work in my favor at some point as all the "crying wolf" of others might make them hesitant to come out next time.