I suspect your display calibration is significantly out of true.
In both your screenshots the grey text is perfectly legible to me, on a fully calibrated machine, and in fact check out to be almost identical values as I see when I switch that setting on, as measured by the Digital Colour Meter in Utilities. That means the data being sent to your display is, near enough, correct.
But for anyone confused as to why your two screenshots look identical to anyone else viewing them here - the 'Increased contrast' setting will actually change the colours transmitted to your screen in a measurable way. The 'Display contrast' slider, however, is calculated "live" & therefore doesn't show in screenshots.
As first steps towards a 'fix' go into System Prefs > Displays > Colour & test any/all other profiles you can see in the list. Must of them will look seven kinds of horrible, as they're not meant for your screen, but one or two might incidentally improve that legibility.
If that's the case, then try a manual calibration.
To the right of that Control Panel is a Calibrate… button. Hold Opt ⌥ then click it. This will present a more advanced calibration set than without opt.
Work your way through it, following the on-screen instructions.
Because it relies on you to guess a bit as to which is quite right, sometimes it's best to go through it a few times, saving as a new name each time & then see which you prefer.
This is not the perfect way to calibrate a screen - for that you need a hardware colorimeter, $£€ 70–300 or more - but it ought to allow enough play in the interpretation to lift in some additional contrast at higher RGB levels.