If you open the footprint chooser, then it tells you in the title bar how many footprints it can find.
KiCad-nightly (V8.0.0~rc2) finds 13481 of them:
(*Which is pretty close to what you found
).
KiCad supports integration with an external database, and I believe this is a better method. It lets everybody use their own database back end (Especially bigger companies likely already have a database with their own preferred parts, including ordering info, etc. So this part is not “ignored”, it’s just implemented in a different way then you expect.
KiCad does have several importers (schematic, PCB, projects, or libraries only) and these are exapanding. In KiCad-Nightly I see: Altium, Eagle, Cadstar, EasyEDA, GEDA). I do not see PADS yet. Maybe you can convert your own libraries via one of those formats. Or else, if you are in to some programming, you can start on creating a converter for PADS libraries :).
As for the searching part…
KiCad does have an integrated search function. Just start typing text in either the symbol or footprint browsers. It’s functionality seems to have degraded a bit recently. I discovered this when test-driving KiCad-Nightly V8.0.0~rc2 and created a topic for this (both on this forum and on gitlab).
(It has received 8 upvotes in just 3 days, and that is quite a lot for such a short time.)