OK... Solved this. The situation was part of a greater problem that happened when I updated from macOS Sequoia to Tahoe.
In addition to disappearing pinned tabs, any new Safari profiles I created would disappear after I quit and relaunched Safari. And profiles I created on my iPhone and iPad (which synced with each other) would not appear on the macOS Safari.
I went to delete the ~Library/Containers/Safari folder as suggested elsewhere. And found TWO Safari folders. Weird that two sub-folders of the same name could co-exist in the same directory. Anyway, I deleted both of them.
After that, everything worked as expected. Profiles synced. Pinned tabs persisted through restarts of Safari. Hallelujah!