These already exist. In Louisiana alone, there were 10 charity hospitals up until a few years ago when their operations were sold to private partners. A few closed, a few remained clinics, but I would wager to say that no significant amount of people in LA is >45 minutes in a car from one of these centers- New Orleans, Houma, Bogaloosa, Baton Rouge, Monroe, Shreveport, Alexandria, Lafayette, Lake Charles. I don't think expansion of the system offers much except for significantly greater cost.

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where healthcare professionals can be required to work 1 day a week in order to “forgive” some their educational debt.

Hard pass. Most of these are in academic centers and don't have difficulty filling staff positions to begin with. There is also already a Public Service Loan Forgiveness program which is somewhat in limbo, but states if you make 10 years of payments (graded by income on a 20 or 25 year payback period) while working for a nonprofit organization, the rest of your debt is forgiven. Usually equates to 3-6 years of residency/Fellowship + a few years more. There was a lot of outcry that physicians would be getting $200,000 tax-free gifts for jobs they would've taken anyway for only 4-7 years of "real world" (not training) work with salaries in the $200,000-400,000 being quite common.

But how about this: let's pretend these centers don't exist and you're facing the scenario where no where in LA offers free care for the indigent. Government is wildly inefficient at running hospitals and clinics. Louisiana just got out of it because of their inability to not lose their arse consistently.
So how about an alternative where the government decides these people need care, allow local providers to sign up to provide it for:
1) the tax break alone of being able to write off these visits
2) some amount of compensation, be it $x/visit or $y/annually if you take on z number of patients
3) loan forgiveness of $X if you enroll in this program where you sign up uninsured and provide care for them in your clinic just like other patients for y years, with a minimum of this many and a maximum set by you

#2 is essentially Medicaid expansion, but there were too many people who needed it driving reimbursement down so far that no provider around most of LA could accept them and keep the lights on.