I don't like how people use the term "stat checking" for the simpler melee champions they don't like, while marksmen are generally the ranged overlords of stat checking.
Ashe, Quinn, and Teemo are the only ones that somewhat deviate from this, regardless of their current strength. The former has a wealth of utility, while the latter two have absurd map control.
All of the others just turn into generic auto-attacking death-dealers, that kill everyone if they're not killed fast enough. They offer nothing to a team outside of damage, so it basically comes down to which one does the highest damage the most reliably. I feel the game would be in a much better state if Riot didn't balance the game around these kind of marksmen.