You really don't get it - and I said $5k, not $5/hr.
You can't live on FT $2.50/hr. Can't be done. In many places, you could in fact survive on $5/hr. Barely.
The unemployed are not unemployed because of minimum wage, they are unemployed because even at 'dirt cheap' no one wants to hire more workers. And definitely no one wants the quality of worker that is actually worth $2.50/hr.
You're concentrating on the case of a single individual, and I get that - yes, if you have a choice to make $80-100 this week, or not, and you need to eat, then you are better off taking the job. But after three weeks like that, when the rent is due, you'll be just as far ahead not bothering to work, because you don't stand a chance.
Get your head out of your ass and realize that minimum wage, at the levels you have, is meaningless in terms of unemployment, inflation, etc. Eliminating it won't make any difference, because you will not create a functioning sub-today's-minimum-wage labour market. So you're busy arguing about something that doesn't have any meaning.