Taking away a low-skilled worker's best bargaining chip (the willingness to work for less money), is generally not a good way to support their ability to gain employment. It does really help middle class teenagers though.
We've seen the tech giants collude to suppress wages, imagine what fast food companies would do if they realized there was no floor to wages.
Finally, I flatly refuse to accept that people are undeserving of a minimal standard of comfort. Just because somebody doesn't know how to code, or hold other marketable skills does not mean we should not value them and the quality of their existence.
No matter what your feelings on the matter, sweeping a floor only has a value up to some hourly rate that depends on a particular business' circumstances. Beyond this point, sweeping will be substituted with technology or simply forgone.
Also, there isn't much empirical evidence to support your claim about a race to the bottom (note: almost all jobs already pay over federal minimum and yet there is no "race to the bottom"). It's also funny you would cite tech workers--some of the highest paid people in the world--as your example of a "race to the bottom". Yes collusion can happen between companies--no one thinks markets are perfect--but markets punish this type of behavior over longer time frames. If Google and Apple are colluding and pushing down wages, some other entrepreneur has an opportunity to pay their workers more and steal their talent. This is slightly complicated by the fact that competition is often constrained in other ways (occupational licensing, regulations, intellectual property, etc.), but these are artificial barriers (i.e., government) to market competition.