There may be people using the 6.5 -06 as a target cartridge but they are few and far between.
While 6.5 mm is currently the most popular caliber (I guess 308 is next due to F Class) barrel life for either the 270 Win or a 6.5 based on the 06 round would be dismal. Recoil would be harsh. Target shooters are trying to balance out highest ballistic coefficient, longest barrel life, lowest recoil. If a barrel blank is $300.00 and rebarreling is another $300, you want something that has an accuracy life of more than 800 rounds.
If I'm not shooting, I'm reloading.