if the primary internet router goes offline but the internet isn’t out (ie a router reboot would fix the problem)
Maybe you just need to give it a simple power cycle remotely? There are devices that do that sort of thing, I have a Digital Loggers Web Power Switch Pro that I’ve used on-and-off over the years for this purpose.
https://www.digital-loggers.com/
At one point I had to relocate for half a year while needing to remote access a slightly unstable desktop that wouldn’t always reboot cleanly and get stuck at the BIOS, it sometimes needed a couple of power cycles to come back online. The Power Switch was perfect for that, I’d log into it remotely and power cycle anything that was plugged into it.
It should work for routers too e.g. it can automatically power cycle something plugged into it based on different conditions like maybe it stops responding to pings or whatever. Or I guess if you had multiple IPs / multiple internet connections the switch itself can stay online and accessed remotely without needing to schedule anything automatic.
Pretty sure there are more pro-level (and more expensive) types of devices to do this sort of thing if you look around
OP’s example use case in the post was with the internet still being up. Building off of that yes, I’d log into the power switch remotely via the internet where I can then power cycle anything plugged into it - for me it was just to restart unresponsive desktops or whatever was plugged into it.
But you wouldn’t need internet to power cycle the internet router itself by using scheduled tasks. e.g. the power switch can check that the internet router is responding to pings every x seconds/minutes and power cycle it if stops responding. (it has other checks/conditions it can use besides simple pings)
That said my own equipment rarely/never needs a reboot so in the case my network loses internet access it usually means the internet is actually down, nothing I can do about that aside from maintaining backup internet if I needed.