Can a picture frame be made from wood molding from a DIY/Big Box store?
Short answer: yes.
You can of course make picture frames from just plain flat boards, even 2x material, so mouldings are certainly a workable starting point for a more elaborate surface profile than flat wood.
I want to make a picture frame while avoiding the routing involved to inset the picture, glass, and supporting backer into the back of the frame.
This is actually the core of the question and it's actually easily resolved since you don't need to do any routing. You can use a build-up method instead.
Here's a cross section of some standard frame profiles with a routed groove along the back:
Shouldn't be hard too see that the top portion, the wide part, could be a bit of standard moulding from the DIY or big-box. The rabbet/rebate can simply be formed by glueing on a second, narrower, piece of wood.
So here's the built-up equivalent:
As you can see this is really very simple and easy so I don't know why it isn't more commonly done by early or beginner woodworkers who don't have a router or other means to easily form the groove, but I suspect it's just that people don't think of it themselves. I know I hadn't, I read of this method in a book and it was an "Ah-hah!" moment because after you see it it does seem head-smackingly obvious.