We wouldn’t need all the infrastructure to ensure only qualified people get welfare. I’ve heard this would make UBI cheaper than what we do now, considering that it would be essentially taxed away from rich people as you say.
Do you have compelling examples on non-means-tested benefits remaining non-means-tested indefinitely? This doesn't seem to be what happens in the UK. Some government invariably either wants to reduce its expenditure or be seen to be "tough on scroungers" and the start means testing.