Does Kodi serve up files? e.g. Can I install Kodi on my PC with all my media, then watch that stuff on my phone when I'm away from the house, or grant a friend access to my library?

That's what Plex does: it has server apps and client apps.

Plex can act as a 'head unit' and do format transform and metadata management. Then stream it to a secondary plex client. KODI does not do that. That is the one killer feature Plex has over KODI. In all ways KODI is better except in that use case. Picking data from a CIFS is basic XBMC/KODI/Plex functionality and has been in there for a long time going back to the original xbox days. Jellyfin is similar with its ability to transcode and stream that to a client. In some cases they let you stream it thru a web client (which is kinda cool).

It is a nice feature for low bandwidth applications. Say a VPN to your phone, or a friends house who has crappy internet. If I remember correctly there were a lot of clients also for TV's which have absolute rubbish CPU power and no local storage (for holding metadata). Also the centrally managed metadata is nice when you have more than one client. You can get the same effect with KODI and using a DB like mysql or mariadb. But it is sort of finicky to setup correctly.

I personally use KODI as I do not need that particular streaming feature. Also it is broken with ISO's which is one of my major use cases.