" I will like us to meet, OZONE, whose job is to make sure that we don't experience this hotness"

I think you're not completely sure about how the "hotness" comes about. Ozone helps absorb UV radiation. But UV radiation does not make us feel hotter. Actually, the rise in temperatures is due mainly to infrared radiation.

UV radiation destroys cells by breaking their DNA apart. When it does, and the cell realizes it can't correct for all the mistakes, it commits suicide, so to speak. In that sense, UV radiation is a huge deal on its own. But it is not a main contributor to the heat problem.

Infrared radiation is responsible for the overheating, although UV radiation also comes into play when the atmosphere fills with CO2 and scattering agents that transform UV radiation into IR radiation via, well, scattering.

This is the Relative Spectral Irradiance. It is simply a normalized version of a spectral irradiance plot, which tells you how intense is the emission for a specific kind of radiation frequency. As you can see, the UV contribution is small, compared to the IR contribution. Notice the plot is in a logarithmic scale.

tl;dr: ozone's job is not protecting against heating, but rather protecting your body cells from killing themselves. However, it also protects UV radiation that might turn into IR radiation under the ozone layer due to scattering.