Looking at https://zoom.earth/maps/pressure/
Given the much lower temperatures at the poles, I would expect the pressure to be (much) higher.
I'm reading here and there that air pressure at the antarctic is low because of its high altitude, but these maps show (I presume?) MSLP?
The ITCZ is an interesting case to use here! You're right that it's the thermal equator and has low pressure, but you've gotta consider convection and wind direction too (i.e. the whole Hadley cell). Convection (caused by solar heating) causes low pressure too, and pressure is often relative.
When you're thinking about stuff on a global scale you've always gotta consider the global atmospheric circulation
There's a lot of good explanation in the link I sent - it's tricky trying to consider all the variables together, but I would say that variations in density (latitudinally, at least - unless you want to start talking about hydrostatic balance!) doesn't account for the variation in pressure or temperature.