Not good IMO, not only all the added weight, but not the best idea to have all your equipment in one easily lost item like that, all your eggs in one basket so to speak. My setup is to have my weapon be just my weapon, with no more attachments than truly necessary, alice gear for general equipment, then a backup of essentials, like another knife and a leg holster attached to my pants belt along with other items stashed here and there. So worst case scenario, even with the loss of weapon and alice gear, I would still have a good knife, essential items, and a .45 pistol. "Dressing in layers" applies to more than keeping warm.
For a "one grab" setup as you're going out the door, just put it in a bugout bag. But in a zombie apocalypse, I would assume that my AO would be the area around my house, and that it would not be a short-term event, so I would want to stay equipped as this is a wartime scenario rather than a flash event like a riot. I also live pretty far out in the country, so would not find myself necessarily overwhelmed by a zombie mob. If I had to leave for some reason, it would not be very difficult or time consuming to properly equip and supply for the trip. Getting my wife properly equipped and out the door in a reasonable amount of time is whole problem in itself though.
A shotgun is a pretty good close range defensive weapon, but in a zombie scenario, I'd still prefer to keep my distance. According to movie zombie lore, a head shot takes them down permanently (at least for my generation's movie zombie, things may have changed nowadays). So my weapon of choice would be a scoped M1A (M14), or an M4 if I wanted to save on weight/space for some reason. Of course in this scenario, my philosophy is that a good offense is better than a good defense, and I would prefer to clear my AO of all zombies, rather than keep dealing with the attacks and having to maintain a maximum level of vigilance 24/7
