My point was that a grant is not "exclusive" if you are required to share it.

The patent system grants a limited monopoly in exchange for complete and enabling disclosure of an invention. It is this disclosure that promotes the useful arts, not the use of the underlying invention. In other words, patent policy does not care whether an inventor actually does anything with his patent. By the time the patent issues, the inventor has already paid his side of the quid-pro-quo.

The theory upon which this trade-off is justified is that if we don't give inventors an incentive to create and disclose their inventions, they will not do so. The limited monopoly purports to serve as this incentive under our current system. Other types of incentives have been suggested, such as prizes, research grants, subsidies, or tax deductions for R&D, but each comes fraught with its own suite of complications.

(Also, I mentioned the First Amendment argument because it comes up in law school discussions on this topic, not because I think it makes sense to me.)

I disagree. A compulsory license would contravene Article I and possibly implicate the First Amendment. It is possible that one may pursue a patent not because they wish to produce a product, but because they wish to prevent others from doing so.

For example, let's say you've patented Method A of processing widgets. Now say you discover Method B, which is less efficient but still viable. You might patent Method B even though you have no interest in using it, just to stop your competitors from moving in an using Method B.

Alternatively, imagine that you hate a particular technology, but you happen to know a lot about it. You could patent a new invention in that technology area just to stop others from using it. In this case, mandating that you license the patent could also be considered an abridgment of your freedom of speech. Of course, it be in the public domain once your patent expires, but for a little while you could make your statement.

References:
US Constitution, Article I, Section 8, Clause 8:
"To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries;" (emphasis added)
First Amendment, in relevant part:
"Congress shall make no law...abridging the freedom of speech..."