It may be your PC that is putting it to sleep as part of the energy savings mode as it is just another HHD. Check your PC settings.

Would something like Dashlane or LastPass (password managers) do an auto login from your browser?

Connect a PC direct to the modem with an Ethernet cable bypassing the router. See if you get the same issue. If so the issue is with the modem or your ISP. If not, try a factory reset and reenter...

Need to isolate the issue a bit. Is it on the wired connection or on WIFI and if WIFI, is it 2.4 or GHz band?

May be channel interference so try a different WIFI channel and channel width. Also be sure the other router is not on if the SSIDs are the same.

@Juliuss It may be that you are not getting a signal from your modem. First try powering the modem, router and PC off and then power the modem up, wait a bit, then the router, wait a bit, and then...