You can do whatever you want while you pay back loans today as long as you keep your contact information current. Just try and move without informing your lenders. That can be treated as a default. And if you can do whatever you want while maintaining a suitable income to make these sort of loan repayments, your experience is an outlier for all but a very small portion of Americans.
This could be run like a regular factory with bookkeeping techniques where labor cost was certainly not zero even though laborers were not getting paid nor free to do anything else but go back to picking cotton.
Labor cost could be reduced to a single line on a spreadsheet like any other business.
After the war with slaves now being paid and free to work elsewhere if they could manage it, this would be offset by the workers paying their own upkeep after the slaveowner had become the employer.
There would have been pressure to keep labor cost from escalating at all if possible.
So things could go forward with as little change in financial risk as could be.
A lot of companies today still have the same type structure as these places had after the war, which can be not much different than the structure they had before the war, and Wall Street is still in the same place it was before the war . . .
A big part of that was the increased productivity of freed laborers being compensated by an increased cost of reproduction.
Having said that, equating all slavery to chattel slavery and thus denying that other slavery-like arrangements have existed and continue to exist today is not only a misleadingly US-centric perspective, it also removes the ability to criticize and try to reform social arrangements that severely restrict freedom in subtler and more nuanced ways.
It is not dissimilar from undischargible debt. In fact, the historians and anthropologists writing about these societies have made the link themselves.