You could take the kids out and about after work (you said around 3ish? Which is a perfect time for after school activities). They could do a lot of their non-independent work on the weekends.
I guess it would depend on the neighborhood and all as to whether I'd be paranoid about cps.
My single working mom let me unschool 8th grade (for which I am still grateful; I really needed the decompression at the time). I was home alone all day and loved it. I read comic books and watched x-files those were the days and FWIW, I was a "latchkey" (tho I'm not sure I like that term) kid before that and after and never felt neglected or scared or anything of the sort. I liked coming home from school and having the house to myself, getting a snack and watching after-school shows or reading *shrug* I think it helped me be independent/self-sufficient, and having my parents trust that I was capable of caring for myself was a big confidence booster.
About the hours, yes 30 seems a bit much. But if you went to work early and they slept late, that would really help a lot. Also, could they buddy up with anybody for one or maybe two days? So they'd only have a few days a week on their own? Also, I think it would depend for me on whether this was a temporary or long-term decision. Will the 10 yr old be doing this for the next 8 years? Or could you get an extra job during the summer and squirrel away some $ so that the rest of the year you could work less hours?
If you were extremely concerned, would it be possible to wait til the oldest is 16 and able to drive? That would probably change the situation a lot. Or possibly go into work after lunch and stay later instead? That would seem more "acceptable" to outsiders, I think, since a lot of kids are home alone in the afternoon.
Some homeschoolers are never home; some are home a lot. We are home-y homeschoolers. We do no more than one "class" outside of home a week, and about the same for playdates. Honestly, I would feel really run down and unable to keep up with our academics and home life if we were out every single day. That's just me and my family, though. We are really happy with the balance that we have.
We must remember that the OP's kids have requested this. They are on board. Because it is their own desire, they will probably be extremely cooperative to make any kind of situation work in which they are able to stay together and continue homeschooling.
If I were to find myself a single mom (again), I would still bend hell and high water to make homeschooling work for my family. Even if that meant working nights while the oldest held down the spot or taking a pay cut for a job that would work out better or even working 12 hour days 3 days a week instead of a regular schedule. It is tough for single mommas, and no the situation will not always be perfect.
OP, kudos to you for looking for solutions to work this out for your family you obviously have a lot of forethought and really care a lot about how strongly your kids feel about this or you wouldn't even be considering it.