If you want better suggestions I'd suggest post a picture.
My favorite screen system is made by this company.
Once installed the screws do not show, if you need to repair a screen just pop off the cover and remove the spline and install new screen.
There's also at least two companys that custom make window system with a film stong enought to withstand a golf ball strike. It's made in sections so you could open them up all the way from the floor to the ceiling and have just screen.
Now you have a 4 season room or can close it if it's raining.