Among the caching nodes in Houdini (Filecache, Labs Filecache) is there one that can be set to "Load from disk" for any frame, but it will also auto-create/save a file if there isn't one for the given frame (without human intervention)?

The original File Cache node can do this, very useful if you want to spit out a bunch of files from a for loop. If you have the Asset Definition Toolbar set to be visible in your parameter pane, you can just switch to the older version. The preference can be changed here:
File > Asset Manager > Configuration > Asset Definition Toolbar:

Thanks. I switched to the old filecache (without the ::2.0 suffix), but what are the settings that would do it? Under Advanced Tab->Load->Missing Frame, the only options are "No Geometry" or "Report Error", there's no "Save" option. I tried checking/unchecking "Load from Disk", I switched the File Mode to Automatic/Read/Write. But I always have to push a button to actually save a missing frame: either the "Save to Disk" button or the "Reload Geometry" button. Can filecache save geo automatically, if a frame is missing?

There is no Advanced Tab in the original File Cache UI, that still sounds like the new SOP. Here is a screenshot. File Mode set to Automatic will automatically write files to disk.

Thanks, I got it to work. Why was this removed from the new filecache? It's very useful.