Frankly, Nozick is no longer a very good representation of modern libertarian theory. In fact, that’s in large part thanks to other libertarian’s rebuttals of his arguments. Arguments in support of voluntary slavery in libertarian theory are the minority. The only living published proponent of whom I’m aware, Walter Block, admits as much:

Now, as it happens, the case for legalizing voluntary slave contracts is held by a minority of libertarians. Indeed, until recently, only Nozick and I held this position, and, according to my friend and colleague David Gordon, Nozick had renounced this viewpoint before his death.

Block, in rejecting inalienable rights, is hardly representative of libertarianism today. Of course, that’s really beside the point. Those in support of voluntary slavery aren’t your audience. Why would they care that the employment contract is similar to self-sale when they support self-sale? Instead, we would ask ourselves: what of the majority of libertarians that reject voluntary slavery yet support the employment contract, as most free-market libertarians do? Are they being hypocrites or, at the very least, inconsistent? To answer that question, we have to understand why they reject voluntary slavery.

One of the most influential thinkers in modern libertarian contract theory is Murray Rothbard. In Ch. 19 of The Ethics of Liberty, he lays out his title-transfer theory of contract. Stephan Kinsella has an analysis and elaboration of this theory here. In that analysis, he draws a distinction between two types of contract enforcement: specific performance (making someone do what they promised to do) or restitution by way of alienable property (paying back the property transferred on condition that the person would perform). For the Rothbardian, this is key. In other words, the Rothbardian would claim that there is another key distinction that you haven’t addressed between the voluntary self-sale contract and the modern employment contract: what counts as legitimate enforcement. [I don't intend to defend Rothbard's theory of contract but only put it out there because it is so influential in the larger capitalist libertarian think tanks like the Mises Institute etc.]

You state in PCE, “the only basic difference is the duration and extent of the two voluntary contracts.” But here the Rothbardian would say,

Not so fast. There is another difference: the type of enforcement that is permitted. Voluntary self-sale contracts, as you describe them, are likely, if they deserve the title “slavery” or “sale”, enforceable by way of specific performance. Without that right to force someone to do something, on what grounds are we to think of it as being anything like slavery, including the “voluntary” kind we find in our history? Those contracts were onerous because once entered into, the person could be forced back if they ever wanted out. It’s only voluntary at the point the contract is made but why should we not care if it’s voluntary at each moment thereafter? If someone where to design a short duration version of such contracts, say for one hour, and the enforcement could be by specific performance, then, yes, you could say that it is just a shorter version of the self-sale contract. But we don’t claim that it should be enforced that way, and that makes all the difference. If the contract is enforced simply by returning the money paid for the hour promised but not performed (assuming it was even paid up front), or simply not paying for the hour and ending the relationship, then it is absurd to make the comparison of employment to self-rental.

Here was how you posed the problem:

If historical slavery was wrong because it was involuntary, then the current system of voluntarily renting other people would be on a different moral footing—but then so would a system of voluntary contractual ownership of other people. If, on the other hand, slavery was wrong—regardless of it being voluntary or involuntary—because it treated persons as things, then the whole current system of renting other people in the employment contract is called into question.

I imagine the following response from a Rothbardian:

You would only be treating persons as things if you used forced to make them do something, i.e. if you enforced the contract as if you owned the person for the duration; but that’s not how a (Rothbardian) libertarian theory of contract works. So we can escape this dilemma by saying that historical slavery was wrong – regardless of it being voluntary or involuntary – because of the force used to back bothwithout that force being a response to aggression. Since that doesn’t apply to employment as we know it, the employment contract is not called into question.

How would you respond to this argument? Has the Rothbardian blocked the way? If their theory of contracts holds, aren’t they being consistent in rejecting voluntary slavery and and accepting non-performance-enforced employment contracts?

Now, even if the Rothbardian succeeds at blocking the characterization of the employment contract as rental just as such, that doesn’t mean that the other leg of your argument – that of the labor theory of property and the legal interpretation of the contract as a rental in deciding the residual claimant – is blocked. In fact, the Rothbardian – by agreeing that labor is not actually transferred to the employer (if it had been, specific performance could be defended) – provides the very ammunition needed to carry your argument through on their own terms. To the extent that capitalism acts as if the contract were a rental, I think you have been let in the backdoor, even if they think they’ve shut you out of the front. The fact that the only way the Rothbardian can classify work is as money transferred on condition of performance (but without enforcement or ownership of said performance) leaves them with little ground to object to labor’s participation in the residual or in the governance of the firm.

As always, I look forward to your insight. I have a vague memory that you might have addressed this move in previous correspondence (or perhaps in published form) but I wasn’t able to find it. I apologize if I’m asking you to repeat yourself.