I own one Axis. Right now, it's a 223. It is stock and shoots just fine. In order for it to be a cheap rifle, Savage put on cheap plastic stocks and so-so triggers. If it is shooting a 1 inch or less, I wouldn't do anything to it. One could make the argument that with a crappy trigger, one learns trigger control. Especially those guys who have had the misfortune of shooting off the rack military rifles from their time in service.
In my case, I look upon some budget rifles as opportunities. They are the modern day equivalent of the hot rods of my youth. I have a chassis that is in need of improvement. When I was younger, I would go to auto supply stores or junkyards to get parts. These days, I get the same enjoyment from taking cheap or beat up rifles and fixing them up.
Is it economical? No. At least, not in the sense that for a little bit more I could buy a Tikka, which will shoot rings around most of the other rifles on the market, regardless of price point.
But here's the big thing - it's fun to restock, or rebarrel, or swap out a crappy trigger for a better one. It's great to mount a good scope on it too. I take the whole shootin' match, if you'll pardon the pun, to the range for a work out. Once things have been checked, I take them hunting. That's entertainment for me. Not everyone can do these things, so I understand the cynicism when I say something like,
"I'm gonna buy a Shilen select match barrel and spin it onto this Axis! I'm gonna replace the trigger too."
This is a picture of the old gauge I use for trigger pull. I check it once a year at the base to make sure it's still accurate, but it's always spot on. I just checked my Axis pull. 5lb. Lots of drag and gritty, but it gets the job done.
People ask me why. My answer is, "That's what I like to do."
If you have skills in this area, it would be fun to involve your child in the build. Make memories.
It's really no different than rebuilding an old Chevy.