If the US had an actual safety net, those laid off workers wouldn't be so bad off either.
In fact, I'd argue that even then we are better off. Say you have 20 low-paying jobs, the sort which means you are still eligible for food stamps and other tax-payer funded assistance. All 20 get laid off. 13 find full time work at the new minimum wage, so they no longer need assistence. Four find part-time work, making nearly what they did before with fewer hours. They still need assistance, but slightly less since they don't need as much child care. One makes less money. The other two are unemployed forever, requiring assistance. I'm gonna guess this is a win in the end.
Actually ending up with a positive outcome requires more reform than just minimum wage - which may actually be negative (on its own, and short term).
There has to be good unemployment benefits for all, you have to have good employment security so hours can't be cut and so on and so forth. The idea is to have such a strong labor collective that driving bad employers out of business is a sustainable strategy. Organization level needs to be pretty high, say at least 50% for unions to have any real power here.
In this case, these shops are located in a sort-of marketplace, where there are dozens of shops selling very similar goods. They all effectively share the same customer base and they are all barely functioning, and are only doing that thanks to avoiding all sorts of laws. If they started paying minimum wage, some of them would close, which would increase profits for the rest, allowing them to stay in business. From the workers' perspective though, it would create a bimodal distribution where employees of surviving shops are better off, while the rest is fired. I'm not sure if that's a positive outcome.