Nothing wrong with what you suggest (my first target rifle was built this way, using a long action Winchester M70 Ranger in .30-06 I bought from Canadian Tire. I still have it and if you come out to an RNBRA match you can shoot it - for example this Sunday at St. Antoine). I'll shoot quite well and it should be able to take you to the very top, or darn near. It might not save you much money over building it with a Barnard action - everything you buy other than the action itself will be the same price, regardless of which action you put it on. The stock. The barrel. The chambering job. The sights. The ammo. Etc.
A Barnard S with trigger is indicated at $1300 on the Hirsch Precision website. I don't know what one can find a bare 700 action for these days ($400? $500?). To this you'll add a trigger ($150? $300? I'm dated, I don't really know). "Converting to single shot" could be as simple as just not using the magazine, so this could be as little as $0. A Rem 700 *might* (then again might not) need to be accurized - perhaps a bit of lathe work on the action face, maybe the lug abutments, maybe the bolt lugs. Maybe lap the lugs. (call the accurizing somewhere from $0 to $200).
Using an off the shelf factory action will be cheaper than a good custom action such as the Barnard. It's just that you should be aware of *how much* the difference is, so you can decide where/if/how you can best save on your costs.
Another thing you can do is to buy an older-generation target rifle and build on that; something like a Musgrave or perhaps a Paramount or RPA. I'd avoid a Swing (pretty fussy and parts can be hard to find) and a Sportco (they can be made to shoot great, but they really are "cheap" and you might feel like you've over-scrimped). Don't get a Lee Enfield (they're too floppy for modern high performance target ammo).
The going rate for a decent used target rifle is on the order of $1000. This would include a good action, a medium or maybe good trigger, a good stock, a bedding job of unknown quality, a good barrel but realistically speaking assume it is shot out and needs to be replaced. Usually you can make a used-looking but good-as-new rifle by simply re-bedding it and installing a new barrel.