Simple solution would be a RPI or router that connects back to your home with wireguard/openvpn so you have a device that you can ping
Alternatively - if your hosting a public webpage - have a cronsctipt run every 2mins that curl PUSH’s to your webpage as a check-in mechinisum. After 6mins - no checkin assume its offline?
We do something like this at scale (about 80k devices) and use a combination or reverse tunnel’s and push notifications. These probes allow tunneling backwards (when enabled) and also do some other local device monitoring. All running on RPI type devices with local wan or 4g
edit: assuming your internet route has a public ip (not behind nat) you could whitelist your own IP and allow it to be pingable? That ofc depends on:
a) having a router that supports that level of custimization
b) having a public ip
Pinging the router would indeed be easier if you have access to public IP (static or dhcp with DDNS). Going the Pi route with WireGuard or another means of “checking in” would be the next best thing.
My comment RE the tunnelling likely doesn’t apply to your situation - was just context that using a Pi would open avenues to other things if you later needed them.