No one rifle can cover the extensive spectrum of varmint shooting - from high volume moderate range, moderate volume ranges up to 400 - 600, then really long ranges for bigger varmints.
.20 Practical - pros: lots of cheap .223 brass, small powder charges, cheap zippy bullets of relatively hi BC for weight. 3600 fps plus with 40 Vmax's., long barrel life.
cons; brass needs continual trimming, .20 Tac with sharper shoulder might help
.22-.250 - must have 8 or faster twist for 75 & heavier bullets - pros: relatively small powder charges for increased ballistic performance with 75 & heavier bullets, .224 bullets less expensive than 6mm bullets. light recoil. Surprisingly good long range performance for a .224. Swats down small rodents with authority & drops yotes fast. 75 H ELDM bullet of choice, good expansion, low form factor & hi velocity, used with RL16, 4350's, PP4000MR & some others.
cons: brass needs continual trimming, .22CM might fix this & give higher velocities.
6mm AI - 8-10 twist depending - more energy for bigger varmints at longer ranges. The 87 VMax works real good and will give good accuracy with 8 twist. The 95 Berger VLD is my choice for long range, great combo of low FF & hi velocity. I started out with an 8 twist that I shot out, then re-barreled to a 10 twist that limited me to 87 VMax's, next barrel will be an 8 twist. Minimum trimming. Wide powder selection, uses same powders as .22-.250 with 87's. High velocity. Rounds feed perfect in Ruger M77 MKII LA. Will see how 6mm CM compares, a little slower but might catch up in other aspects.
cons; fire forming brass & 6mm brass is sort of scarce. 6mm CM expected to fix brass availability problem.
The 6.5-06 gets some use with 120 H ELDM's.