The thing is that the unnecesaary work isn't "busy work" but is work that primarily delivers value to the employer by winning in zero sum competitions. Advertising, legal services, lots of software development, lobbying, plus all those services that enable high productivity for zero sum work.
The problem is that we are dependent on these zero sum games to regulate our economy and as a result these zero sum games end up consuming all our our productivity gains and making up a progressively larger share of the economy.
Apart from needing those technologies to exist before being able to claim them as reasons there won't be a labor shortage with UBI this is still completely independent of rich people existing today without causing a labor shortage. Perhaps it's a funding source to prove out things like automation by redistributing said wealth but the existence of said wealth in the hands of a few is not prove lack of labor in all is sustainable. To do that you must actually prove the alternative means of production out - regardless if the wealthy exist.