What's wrong with this, honestly? If less desirable jobs go unfilled, I'd expect that people would find ways to mitigate the need for human beings in those jobs. Let's let the market figure out what those jobs are and if we can do such mitigation. That seems preferable to the current situation, where people such as yourself say that we need people to be placed into explicitly undesirable positions.
Let's actually see this problem before we anticipate it.
The problem has always existed. It's the reason that people need to be paid for most jobs, since they won't do them just for fun. If people know that they'll be paid a decent amount for doing any job at all, then there will be a lot of recreational activities which are thinly disguised to look like jobs.
So in your case under a BIG, they would get some money and pursue leisurly activities. This in contrast to someone doing a task that needs to be done (say dispose of garbage), who would get some money from the BIG and a substantially bigger sum in actual wage. The service (getting your garbage disposed) would have to be priced accordingly.
Some activities (such as musicians) would be a gray area, but it's not a problem - people can make music and if someone buys it the musicians get extra income.
If there are services needed which are very unappealing, they will be priced high. Also, the incentive to automate them (and thus reduce human suffering) will be high as well.
Why would we do that? We could instead just sell the "something", which has the same effect, without defeating the point of BIG by complicating the BIG administration.
There's a huge difference between (a) not being interested in doing a job, but doing it because you're getting money for it and (b) not being interested in working and doing it because you're getting money for it.
You're arguing (a), but I'm arguing (b).
I think eliminating the minimum wage is reasonable combined with a guaranteed stipend. However, I think you will find that "undesirable" jobs are (rightfully) paid quite well. Possibly even better than now since this system would better balance the power between employers and employees.
It's particularly silly with the current system that you can legally work for nothing, be a volunteer or an unpaid intern, but you can't work for $1 per hour. If you make low paid work illegal, you make the low paid unemployed.
Perhaps some day all of this "scarcity economics" will be moot, if we could invent the star-trek style replicator, I suspect "work" would move to a volunteer model.