I feel like foregoing the whole PCB would be better, and just wirewrap, or "free-air" solder.

Perhaps.

This has an advantage that the board itself is printed.

After molding and firing (say) 50 of them, those that survive will all look and work about the same. Painting the conductive traces into the printed pathways is an easy thing to get right. And then the parts are soldered on, which is also easy to get right.

The design and the pathways are predefined, and then mechanically copied (printed) over and over.

This reduces the skill required for final assembly.

Wire-wrap and point-to-point methods certainly also work, but they come with increased potential for errors at assembly so getting them right tends to require more skill. Reducing assembly skill is part of how PCBs became commonplace to begin with.

And those other methods still generally want a board of some kind to mount stuff to, anyway, just for practical handling and durability reasons. It might be a perfboard. It might also be a chunk of scrap wood from the shed (we can even add nails and Fahnestock clips to it for fixturing and connectivity!). Whatever it is, it probably still resembles a board.

But with this clay method, the provision of that board is inherent in the process. That has a distinct bit of elegance to it.

(And if we cast all logic and reason aside, then remember: This is supposed to be art. It's OK that different methods of circuit assembly exist, and it's even OK if some or all of them are better in some way.)

How would you handle LQFP or BGA packages?