It's funny that the OP brings this up, because I decided to listen in on our local Skywarn net last week and this is pretty much what I heard during a Severe Thunderstorm Watch. There were some basic storm reports, but nothing reaching severe parameters. Interestingly enough, the watch expired following an afternoon of relatively small shower activity, and then an epic lightning storm with severe warnings arrived around midnight with the Skywarn net being dead quiet as far as I could tell.
As far as the basic chatter during the day, periodically the NCO would break in and inform everyone that Skywarn reports had priority. Those chatting away were reminded to provide breaks in between brief chatter. Apparently, this is only true during standby mode when a watch is present but no warnings are. During LWX CWA warnings it becomes a directed net.
Some of the NWS Skywarn ARS groups IME are very well coordinated and professional. I haven't been involved long enough to share experiences elsewhere, but I will certainly add more input as I travel and experience other CWAs.
Regarding the phones, especially cells, given my experience I've viewed them as tools that can and will fail. I certainly keep the appropriate numbers for Skywarn written down, as well as plugged into my cell, and I possess solid, reliable NWS contacts via phone and email, but I'll always keep a radio handy. Redundancy is a good thing.