It's true that a sufficiently high UBI could serve as an alternative to a $15 minimum wage, but I personally would prefer a combo of both plus a 4-day 32-hour week. Let's pay people more for work, support all unpaid work, and distribute employment and leisure time more equitably.
What the left needs to be honest about in regards to a $15 min wage is that although the overall effects will likely be positive, there will be impacts like reduced hours to compensate. Let's lean into that by leaving the 5-day 40-hour week behind. Every weekend should be 3 days.
It's also likely that a higher min wage will increase automation. Great! Let's do that! But that means less employment for humans. 4-day weeks share the available employment better, and universal basic income makes automation literally work for
UBI makes sure that micro-level responses to a higher minimum wage don't reduce people's total incomes if hours drop, or drop incomes to zero. For that person who can't find a job to pay them $15/hr, they can have $1500/mo vs $0. And unpaid workers would have $1500/mo vs $0 too.
If all we did was redefine full-time as 32 hours, businesses would need to pay more per hour so that employees got paid the same for less time. That would distribute leisure time better and I think raise productivity, but it wouldn't raise total incomes, which need to be raised.