The price depends heavily on your wishes... you can buy a central unit for a few thousand or even a few tens of thousands depending on what is desired...
The cheaper devices usually just cool with a lot of energy consumption... the more expensive ones can be adapted in many ways to the user/house and are more economical.
It is built almost like a controlled residential ventilation system but with larger ducts since larger volumes of air have to be moved.
It cannot be done via controlled residential ventilation because it is not designed for the amounts of condensate and air volumes that occur with active cooling.
No, I only have a split unit too because I was foolish enough to think during construction: "oh, the house won't get warm in summer, there's enough insulation everywhere."
Now, after the first summer where I could hardly or not sleep at all during the few weeks of heat, a split unit came into the bedroom.