I think you too are talking past each other because both of you are focusing only on the money, and not the obligation, and just looking at the money, and who owns it is not how modern finance works.
Modern finance is essentially the Art of I.O.U's.
Before Grimm1's instance of someone else getting hired on, your hypothetical employer has their affairs set up such that expected payroll is met. Payroll is calculated as a function of headcount, and often, the business owner/primary equityholder are the end eater's of any financial shortfalls. They get paid last. Payroll must be paid first, vendors generally fall somewhere in between, and are somewhat more flexible in how you can accommodate them, as long as you make good on it.
Grimm1's scenario would therefore properly play out as the Owner doesn't take home as much (honestly any profit if things have deteriorated to the point you can't meet payroll).
The courts do not look favorably on not paying your people, and in bankruptcy, your payroll obligations take primacy(I think).
You (dataflow) don't necessarily paint the most accurate picture in the sense that the liability does in fact exist in a materially fashion. You can "call it" in the same way that a bank can do a "margin call", which usually requires getting an agent of the court involved.
The sad thing is, legal representation/effective litigation has such a high barrier to entry, that most people don't even realize they can. The other problem is the additional legal costs may jeopardize a business to the point it death spirals, ensuring you never get made whole.
USPS is an interesting case, because I'm not sure what else might be in play due to it's quasi-public nature. A federal institution cannot go into arrears. (Literally, a lot of Federal managers won't even be comfortable letting you do volunteer work in my experience). They have to furlough on budget shortfalls. But USPS is not "Federal" like most Executive agencies, as they are still technically private, or have been operated that way, while also being Constitutionally mandated to exist.
USPS as I recall, is also one of the only employers required to pre-fund their pension liabilities. Unlike everyone else in the private sector who apparently aren't.