That sort of thing seems like a good idea. There are X programs (and presumably Wayland programs) that do similar things (like web browsers stopping things when switching to a different tab or minimising the window). But we could probably do better with different interfaces.
For the moment the Librem 5 seems to be using apps designed to work on PC desktops but at lower resolutions.
Do you know of a good web site explaining how this works? My Android phone used to kill my Jabber client all the time until I got to using Conversations which has a notification all the time which seems to be the way to avoid being killed while running in the background.
It would be nice to be able to reliably run background apps on Android.
Also as an aside Android doesn't appear to reliably kill background processes, it kills them if it thinks that something else needs the resources. Running the Facebook app is one way of triggering Android to kill a bunch of background apps.
My first Android phone was a Sony Ericsson Xperia x10i. With that phone I could go to sleep while playing music from the SD card and wake up 8 hours later with plenty of battery left. The same phone however would run out faster if doing stuff over Wifi or using GPS. It was mostly a matter of how much power different things took.
One of the things I want to do on my Librem5 is monitor my servers, so that will involve polling things every few minutes. PowerTop says that I can save power by changing the polling for USB, but that changes Wifi ping times from ~1ms to ~350ms. Eventually I'll probably try experimenting with that to get an option with a 10ms ping time that still saves some power.
All programs on the Librem5 appear to be in Debian packages and most of them seem to be identical to the ones in Debian. The document you cited has procedures for getting packages in PureOS independently of Debian, but it seems that most of it will be stock Debian. For my development it seems easier to just upload to Debian and wait for the next Debian release for it to be officially part of PureOS, I'll setup my own apt repository for the things I do and publish the URL for anyone who's interested.
As for apps being suspended, most apps are suspended when there's nothing to do. If a graphical application is minimised so it doesn't have to redraw the screen then it should either be doing nothing or occasionally polling a server if that's it's design.
Web browsers are an interesting corner case as web sites often have JavaScript that wants to run all the time and there's some trade-off between doing what the web site wants and saving CPU/energy. But that's probably not going to be an OS issue for PureOS but an Epiphany browser issue.
It is thick for many reasons, having separate chips for security means a bigger motherboard, having a socket for a security chip, etc. But the battery is really big, much bigger than is common for modern phones.
The Librem5 is twice as thick as any other phone I have owned in the last 10 years. In general I agree that thicker and heavier phones would be good but in this case they seem to have already got to the upper limits of what's considered acceptable nowadays.
For the moment the Librem 5 seems to be using apps designed to work on PC desktops but at lower resolutions.