Well, if no one would work the unrewarding jobs, and they were needed. The wages for these jobs would go up. As Bertrand Russell said there's two types of jobs. "The first kind is unpleasant and ill paid; the second is pleasant and highly paid". Reversing this might not be the worst thing in the world.

And/or they'll be automated (or at least sufficiently automated).

Unrewarding jobs would pay more, in some combination of money, perks, social status... To the degree the last held true, they'd become more rewarding. To the degree the first held true, there would be increased pressure to automate.

Unrewarding jobs in an ideal world would pay more.

Let's think this through ... how much would they pay more ? Well, for most unrewarding jobs I think you'll find that labor costs dominate. So any wage rise needs to be reflected in the cost of those services ... which results in a need for more money (since cost rises in basic goods will disproportionally affect poor people, resulting in the need for further basic income rises)

So it's obvious there is a feedback loop here. Basic income -> price hikes -> more basic income -> more price hikes ... The question is what multiplier we have between the steps. If that number is << 1 (much smaller than one) ... then no problem. If it's anywhere near 1, basic income will result in total devastation of the economy.

There are jobs that would be affected more than others. Elderly care, for example, is a very dirty, very unrewarding job. With basic income it'll become completely impossible to pay for. So that looks to me a very basic tradeoff : you can have one of these : elderly

And that is a consequence that's very unlikely to be appreciated by either rich or poor.

So you're suggesting that a Basic Income would drive increasing automation in virtuous cycle as labor costs rapidly rise to more than the cost of designing and building the automation?