I think the changes that should be done are not directly connected to competitive, but Valve really needs to attract more people to the game's competitive mode.
Playing the tutorial should be mandatory and unskipable for every new player and two new parts should be added (Medic and some very very basic rocket and sticky jumping)
Give players a Strange weapon or a craft hat whenever they reach a level for the first time in competitive or a random drop every for every 3 gold medals, things like that.
These are things that people would be willing to play for and actually tryhard in the gamemode.
Also, the "obvious" stuff, which Valve that supposedly plays the game a lot hasn't fixed since the launch of the game.
Proper net settings because playing without these (especially on projectile classes) is painful.
Brighter and slightly bigger damage numbers.
FOV locked at 90 for everyone.
All of the above settings locked(?) and on by default.