they can make the FAC intersect the extended centerline anywhere
they want.

In theory, yes.

However, I have called Flight Procedures to ask about approaches
similar to this, and the reason was the one I gave.

There are sometimes obstacles that can be avoided by aiming the FAC
towards the center of the airport, rather than the runway itself, and
the placement of the intersection becomes limited.

Unsatisfying, I know, because of its vagueness.