right. thats the point and how it works.
you monitor your device from outside, if you cannot reach them then you have a problem. device down or internet down. no matter what the reason is, you have to fix it asap.
the master suggestion is not opening ports and use vpn. so if you use vpn you have to open at least one port which you can monitor.... i disable ping for a good reason in my routers, but sure. you can use a
ddns , allow ping from outside and ping the router.
dont know why you want to make it complicate, but this is the normal way to monitor iot devices.. you can use amazon or google cloud for that... i think they offer it also as a service...
OP wants to get a notfication if router (internet) is offline. For what reason do he need (redundant) internet on site for THIS ?
even a backup internet connection wouldnt be enough. you need a redundant power supply and also a backup battery. what happens after your battery dies ? right you need a generator..
where do you start and where do you end overengineering a simple question: how can i get an alert if my router is offline?