most varmint rifles are pretty accurate to begin with.the scope that you put on the rifle might cost as much as the rifle are more unless you are going to have a custom rifle built for you.And the scope will help make it accurate.the caliber you selected,.223 is a pretty good choice.ammo is available just about any place that sells ammo.and is pretty cheap.But if you are serious about it.I would opt for the 22-250,I do not beleive you would be sorry with that choice.
A lot depends on what varmints you are going to shoot,and the distances involved.as the .223 when you wanted to, you could have it rechambered to the .223 ackley improved.that is a sweet round.
and as for your question "your most accurate varmint rifle".Gaaaads..They are all very accurate..and I have many.but am very partial to my little remington 600, is a 4 digit serial number in .222 remington,have done nothing to it,except load quality ammo for it.and all I seem to load is 40 grain bullets.it has the short barrel on it.and is my truck gun, normally when I am off to hunt ,shoot what ever it's next to me by the front seat.it has dumped a lot of freebie coyotes,and mucho P-dogs driving thru towns for the next set up.and I find it a plesure to shoot.no recoil,can see the hits in the scope normally.