Welcome to the community.
I guess the question that I would ask is: Is the expectation that all this would happen only when the phone is in active use or also when it is idle?
The reason I ask is that at some times when the phone is idle it wants to go to sleep - and there is a trade-off in keeping it awake in order to do things but thereby reducing time between charges.
I think it would be badness to use the red LED in this way because the red LED is used to indicate what’s happening with charging. So are the colours negotiable? 
The blue LED is used for notifications - but that includes quite a few different notifications, not just e.g. a new SMS.
The green LED is used during the boot but once booted I don’t think it is used for anything else.
I’m a bit unconvinced about the merits of knowing when the phone is connected to a network. I mean unless you are in a flaky signal area, the phone is (more or less) always connected to the mobile network, unless you’ve switched the modem off. Are we talking only about the mobile network or also talking about WiFi?
As to the actual question, maybe here: A way to disable blinking red led? It’s not something that I have tried however. Do you have a Librem 5 yourself to test with? (Note also that even though I have linked you to a topic about disabling the blinking red LED, you can’t do that and the actual post shows someone enabling the green LED.)