Chefs do “robotic jobs” repeatedly creating the same end product. Would you be comfortable if the person who prepared your Michelin course dinner was forced to urinate in a bottle while they did it? Why are we ok with treating human beings like this, simply because we do not romanticize the task of sorting and moving boxes.
Modern American safety net is far from perfect, but it does a pretty good job making sure people don't die from hunger not freeze to death.
There are many things that could be automated right now (for example, in the garments industry) but are not because it's currently cheaper to do them in a third world country by poorly paid workers - but the moment those workers would have to be properly paid, the tasks would go away as then it would be cheaper to automate.
Which means less labor supply, which means higher labor prices, which means greater incentive for people to work (since wages would no longer be suppressed by the "either I take this predatory offer or I starve to death" effect).
> Then the people still working will have higher taxes
Not necessarily. That depends on who and what you tax. A land value tax + a tax on income or wealth after a certain point (say, $1m/year) would readily pay for UBI without putting any tax burden on the working class. The other benefits of LVT (like incentivizing denser urbanization and penalizing land speculation) are nice cherries on top.
...which means higher prices of good and services, which means that the basic income you had previously settled on is no longer sufficient to survive on. How sure are we that there's a stable equilibrium?
Only if automation never happens. Clearly that ain't the case, per this very comment section and the topic thereof.
Indeed, automation's going to continue anyway, because as automation costs fall they'll eventually undercut labor costs - even those artifically suppressed through coercive means, as is currently the case in any (capitalist or otherwise) society lacking a socioeconomic safety net like UBI. So sooner or later something like UBI is necessary either way (that, or making people do pointless busy work, which is both paternalistic and grossly inefficient).
To realize what people are going to do when they don't have to work, look at those in retirement: gardening, choir singing, lotto may be?
But a lot of people if guaranteed a minimal quality of living would just stop working and do things that they like (that don't necessarily make money) and that's the whole point of UBI.