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I do not believe this is the case. Supreme Kai does not govern over every timeline, but the Supreme Kai of Time does, so it really wouldn't make sense for there to be infinite Supreme Kais of Time. If that was the case, you'd think the Supreme Kai of Time from Future Trunks' timeline would be in contact with the Supreme Kai of Time from the main timeline we are working with.
This led to alt timeline Trunks traveling back to the percieved 'main' timeline for help with Goku Black following him to what would be a third timeline overall. (This timeline's Zamasu was erased by Beerus)
Also Mai was made young again in Trunks' timeline despite not having any of the events of the DBS movies with the dragonballs. (She's originally older than Goku)
TL:DR
Timelines seem to contain all universes involved therefore I'm gonna go with a yes to being in charge of all of them.
Actually a special chapter for the Super manga confirmed that the Pilaf gang made their wish to be young during the Android crisis (many suspected this, as it lines up Mai and Trunks's age). However in Future Trunks's timeline the Pilaf gang stole the wish from Piccolo. Which is followed by the Androids arriving and killing Piccolo, making it Future Earth's last wish for Shenron.