Edit: It is definitely above your $20 price range, but an idea for the future.

To this day, I still always carry my Sunwayman V10R that I modded with a neutral white XPL HI (was early generation XPG cool-white stock). I love it, and it’s perfect for me. Definitely look into it as the only “modes” are the brightness increasing or decreasing as you move the variable output ring on the head. The low is insanely low as well.