The construction adhesive will minimize squeaks. If you don't mind a few face nails in the treads you can pin them in place with an 18 ga nailer, then drive screws through the risers into the back of the treads and pocket screws into the underside. You can use pocket screws through the stringers as well. I would use PL on all those joints, but sparingly where it can ooze visibly. You will want some kind of adjustable template to get the length and angles of the treads and risers correct. This is where housed stringers have an advantage as the tread and riser lengths can be looser. You can also install the housed stringers and wedge the treads and risers in place one by one. I know you have rejected that approach but it is easier to achieve good fits. Here's a thread showing a housed winder staircase.
https://sawmillcreek.org/showthread....ancing-winders