weirdly, this is next to impossible in FCP (classic, i don’t know about X)
but you can do it via XMLs, & 3rd party apps
the primary one i know of is from Spherico Film tools.
you export an XML from FCP,
run it thru the Filmtools app, and get an XML back that opens in FCP with text generators over your clips.
RESOLVE (which is free) can also do this.
so you could export an XML from FCP, open it in resolve, and after tweaking all the right settings,
be able to export a file with all the clip names burnt in.
i *think* you can engineer the export to make a bunch of separate clips, all with burn-ins.
i know Resolve is used for data wranglng and that is a a fairly common need in that world.
if this is something you want / need to do regularly,
you might be best of making burn-in versions of your archive clips, BEFORE you start editing
or make burn-in versions and re-connect to them which is a tricky operation, but not impossible.
nick