My suspicions are a ROUTER issue rather than a MODEM issue.
In every single case I have looked at where the MODEM has lost sync, it usually resyncs very quickly with no human intervention.
On the rare occasion that the MODEM hasn't automatically resynced, a MODEM reboot does NOT fix anything, the fault being on BT's side of the modem i.e. an external cabling or (accidental) engineer's physical disconnection either at the cabinet or somewhere along the route from the cabinet to the home.
We have previously read of some BT HomeHubs needing to be disconnected/reconnected. It may just be that one of these "faulty" hubs has been provided & needs replacement.
A MODEM reboot will ALWAYS refresh the ROUTER'S PPP session, but as mentioned by B*cat, it isn't always necessary to reboot the MODEM itself.
A monitored unlocked Huawei HG612 modem is probably going to be the best way of actually seeing/recording what is actually going on.
It may actually identify a specific issue that can be reported as a "fault".