The PSI keyless chuck looks like it would work just fine. I believe all my keyless chucks were right around that price point too, and I have not had any problems with mine.
What size boxes are you thinking of making?
The most common forstner bit that I use is a 2 1/8, which is a very convenient size for me since it is the largest one that I have, and the 2 inch chuck jaws fit into the mortise that this cuts, making bowl mounting a lot easier for me. Other than that, I don't really use any other sizes for my turning. I've thought about larger forstner bits, but there just doesn't seem to be one that has universally great reviews - if you find one though, let me know!
The 2 1/8 forstner bit that I use the most is also the 'wave' style, but by freud:
https://smile.amazon.com/gp/product/B0000AV74R/
I do have a woodcraft set from 1/4 through 2 1/8, and I use those in my hand drills when needed typically.