What a good story! I can see you all bundled up, toddler in tow, struggling to get that real tree, then with toddler and baby, going for the easier option.
I have a similar story. I went with all three of my young children to get a tree, but I was too late! All the local tree suppliers were done for the season. It must have been xmas eve. So I too went to a mall, possibly Target, and got my first artificial tree, which we used for many years. It started shedding just like yours, and some of the lights no longer worked. It definitely looked real though, After thinking I would never get an artificial tree, never ever, I converted. So much easier!
You do have similar tale to tell! I, too, never thought I'd be content with an artificial tree, but that first one was a game-changer.