@Celia boxing the corners is much quicker than adding a gusset.
On some of my bags, rather than trim off the triangle piece created by the boxing out I fix it to the side of the bag, this may help. I usually rivet them in place but that’s only an option if it’s a reasonably firm fabric and you have rivets!
You could consider putting something in the stiffen the base. I would use something like those very flexible cutting mats. They are very easy to trim down and if you tuck them down well they should do the job.