If my understanding is correct then it is the Speedforce itself which chooses who gets to tap its power.
This is a murky topic to discuss as in the past i has been shown that the Force is what enables such monumental power to speedsters like The Flash family, but when the force is removed it takes most of that power away and will leave a Speedster at a level around the speed of sound (Jay and Wally), which suggests the Force is a supplemental agent rather than the font of all Superspeed itself.
Since Quicksilver already has vast Superspeed thanks to his Mutation I don't believe he would be eligible to tap the Speedforce, the Force appears to grant access to normal speedless humans and the likes of Kryptonians, Mutants, and bionically powered speedsters are their own source of power... but the whole subject is open to some interpretation.
The SpeedForce is a sentient power residing just in the DCU, as the flash was cut off from it in Marvel. and you are correct, it selects whom it wants to have the speed power...
Superman and others like him, Mon El. are not able to access and use it, but they are pretty much as fast as any of the flashes, at least before they go "all out"
And Capt Marvel/Ba seem to be about as fast also, due to having speed of "gods"