The liquid should be able to hold less gas at higher temperatures, yes.
What you observe with boiling water though is not solved gases leaving the water but actually water turning into steam. Steam is exactly the same stuff as water, just a lot less dense.
Actually if you increase temperature and pressure enough, there will be a point at which it makes no sense to differentiate between water and steam anymore [1]! Which is a fun thing: you can turn water into steam without boiling it. Raise the temperature and pressure enough while keeping it liquid, reduce temperature while keeping the pressure up and then reduce the pressure as far as you like, as long as it doesn't make the water condensate.
Voilà, steam without boiling!
What is happening there is taking a detour around the critical point instead of crossing the blue line (which is where water boils) in the following graphic[0]:
#/media/File:Phase-diag2.svg){kind=link}