Your concern about UBI giving money to the very rich doesn't make sense. They'll be given $12k, but they're paying, let's say, $500k in taxes. And if taxes need to be raised to cover this, then they'll be paying much more than $12k more to do it. Meanwhile, a poor person would be given $12k, and they're paying $0 in taxes.

Same reason it's nonsense to complain about including rich people in free healthcare, college, etc.

When you try to means test things, you end up needing much more administrative machinery, and you end up with weird discontinuities where it can be better not to work so you don't get cutoff from benefits and end up worse off overall.