Can I really use construction adhesive to attach hardwood stair treads and risers?
Simple answer... NO!!! As someone who's done a lot of hardwood flooring, as well as stairs, adhesives are not enough to hold a stair tread down. Treads should be attached with a combination of adhesive and nails. Nail length being a minimum of twice the thickness of your tread and of 16 gauge or larger.

You can just glue the risers but that's not something I'd personally do either. I'd use adhesive and a few Brad nails ( 18 gauge ) will suffice.

Most construction adhesives dry hard. And over time, simply break their bond to the stringer.