We took the ladder to the back of the house where it would have a flat space of ground and where the roof was only a story and a half up. We bundled the necessary equipment in a sack with a strap, and I climbed the ladder. Looking out onto the roof, I forgot about heights and focused on my goal (which was on the other side of the roof from where I had climbed. I set the sack on the hardened snow and was heartened by how firmly it stayed. I ventured a step on the roof and my foot immediately started to slide away from me, pushing the ice from the roof. I grabbed the sack and came back down.
Connor felt that he had a better shot being more athletic, agile, and (frankly) lighter than I am. However, he thought better of the prospect the moment he got to the top of the ladder, and also decided to come back down. I feel that our prudence was justified.
We had another option left, and that was attaching the dish where we were, near the easternmost corner of the house. I went up the ladder again and took a reading. There are some trees to the north and south, but the app said that the placement was minimally obstructed (as opposed to just "obstructed" in the other location). We put it up there with the cord now coming into the house through our office window.
Sadly, it seems that our app was incorrect as--after 12 hours in its current placement--Starlink said that the location was obscured. Since then, we've almost had worse results; although a factory reset did seem to clear them up a bit. We also discovered that it really didn't like the heavy snowfall that we received for a while today even if it seems to have recovered since then.
At this point we basically need one of three things: a stable ladder that can reach to the top of three stories, a cherry-picker lift (I don't think that a scissor lift would make it onto our deck--especially with the snow, or a professional with the correct equipment. I've also considered the idea of putting it at the top of the garage roof (which should have a clearer line of sight even if it's lower), but I'm concerned about what to do with the cord to get it into the house--although, I suppose that I could just have the router in the garage itself ... I'll have to think about it. All I know is that it is not going to work where it currently resides.