The ITU does have jurisdiction. Since SpaceX (and Amazon) are US companies, they work with the FCC, who then works with the ITU on behalf of SpaceX/Amazon.
How come I can't easily find a registry of the UP/DL/Beacon transmission frequencies for every satellite. Am I just not looking in the right place? I can find various hobbyist's catalogs, but nothing that looks remotely official. Any satellite with downlinks between 30 MHz and 1.2 GHz is potentially useful for helping to calibrate one or more of the radio neutrino experiments I work on.
A good source of downlink information is the FCC International Bureau records of satellite ground stations in the US.
Also note that this shows licences, not actual use. You can have a satellite licenced to a particular frequency but not using it.