It doesn't matter whether the overwhelming majority of people would be willing to clean toilets for however much money: what matters is whether enough people would that the toilets get cleaned. From what I can tell, the answer to that is "yes".
Given instructions, and absent other immediate obligations, I would do so much helping out, wherever I happened to be at the time. (The only reason I don't now is because I don't understand most jobs – my meddling could do more harm than good –, and they won't let me do jobs in my area of expertise.) I'm not unusual in this regard: perhaps I'm unusual in that I'll do this unprompted, but if it's a societal expectation that people clean up after themselves, and leave things in a slightly better state than they found them, people generally do it.
The problem is not a lack of workers. The problem is not a lack of things that need doing. The problem is a lack of "jobs". UBI (with the necessary patches to, e.g., prevent bad actors from redirecting all the money) is essentially employing everybody to do what they believe needs doing.
So the question becomes: do you believe direct democracy works at small scales? Your answer appears to be "no".
I have no idea what you're basing "from what I can tell" on. In America there are literally millions of janitorial and cleaning staff. The overwhelming majority of those people would much prefer to e.g. spend time with their family than engage in menial labor to make ends meet. And the marginal utility of money decreases dramatically, at least for most people, once you have enough to comfortably survive and provide for your family indefinitely.
I'm a white-collar knowledge worker. There is no downside to me getting my hands dirty with real work during my thinking time – that's time I can't spend with my family if I want to get anything done, because unstructured social interaction takes up the same space in my brain as the knowledge work does.
And the marginal utility of having cleaner toilet facilities increases dramatically when they tip over the boundary between "clean" and "not-clean". Given an apron and long gloves, I'd happily (most days, at least) clean shit off the wall, or pull pads out the U-bend, if it increased the chance the room was clean when I had to use it.
If I were able to choose whether or not to clean the toilets, and someone kept leaving them in a right state, then after the second-to-fourth time, I'd set aside some time to identify the culprits, and then I'd take measures to ensure they stopped, starting with confronting them over it: first in private (if possible), and subsequently in front of people whose opinions I'd expect them to care about. I might escalate further if it continued without good reason. This is not an option available to cleaning staff on the poverty line, who rarely have the time in their schedule to do this, and might be sacked if they used their breaktime to pull an insubordinate stunt like that. (I would not be capable of working in those conditions, and I expect I'd develop new and fascinating mental health problems if subjected to them.)
There is a major difference between volunteering to do something, and having your ability to live in your home contingent upon you doing it, as regards people's willingness to do various tasks.