Their job description is to keep aircraft apart not allow as many aircraft to fly as possible
Well, not quite, and certainly not in the UK ....
"An ATC service is provided according to the particular circumstances and class of
airspace, for the purpose of:
a) preventing collisions between aircraft in the air;
b) assisting in preventing collisions between aircraft moving on the apron and the
manoeuvring area;
c) assisting in preventing collisions between aircraft and obstructions on the
manoeuvring area;
d) expediting and maintaining an orderly flow of air traffic."
MATS Part 1, page 1