I have never had any problem with stretching when I do them this way, but it was just a thought.

I've sewn unstarched bias strips together with no problem.

Of course, it helps if the fabric is relatively stable to begin with.

Not if the bag is lined.

I do think the best idea is to foundation-piece on thin muslin. The muslin would provide additional stability to the pieced fabric. I would still cut the strips on the straight-of-grain.

The easiest seams are straight-of-grain to straight-of-grain. Sewing straight-of-grain to bias is much trickier. Sewing bias to bias is just about impossible unless you heavily starch the fabric before cutting so the fabric has no chance to stretch out of shape when you sew the seam.

I agree with the idea of a foundation fabric for a base. This is one of those "form and function" things.

Your original question was about why you would cut on the bias. That question was answered.

Now if you want to know how to make this pattern for a purse. That is a different story. This is where function comes into play. Thickness and weight really aren't a problem in this case so sewing strips onto a foundation fabric can be your answer to solving your bias issues. I really am not sure you are going to save that much in fabric in the end (I didn't bother to plot this out and do the math).

Personally, I would go the foundation fabric route. Just my $0.04. (Prices are going up.)

I made handbags for my granddaughters with jelly roll strips. I sewed the strips together, pressed seams to one side, then layed my pattern piece so the strips were at 45 degrees and cut them out, no starch, no backing, had no problem with stretching. Once you get the zipper & lining in there won't be any problem with stretching. Just pin often when inserting your zipper. I have posted this picture before, but you can get the idea.