I do empathise but I also think you are making it a bit bigger than it needs to be.
I'm a single parent to 3. Very little involvement from their father & one DC does not see him at all so I always have her & her activities to deal with. I work f/t.
My kids are early teen / pre teen & all have jobs which they do, but I'm still drowning in housework.
They do many sports so we are literally on the go all weekend (I feel I should reduce this, but they love their sports).
I am at a loss as to how a 3 yo & one adult can generate that much cleaning.
For me, I try and do a version of organised mum tho very much my own version, as in jobs in quick pockets of time eg 15 min ironing, toilets, sinks - all done separately
I find my kids mess it up very easily, and while they will clean, it's not as good as I'd like.
I know that having clutter, which I do in several rooms, is adding to the time to clean, so I'm trying to deal with that.
I can't afford it right now but I think I will aim to get a cleaner eventually. Funnily, I like cleaning, it's just the time element.
For you, I'd do a schedule of certain tasks on certain days & stick to that. Limit toddler mess to certain areas.