jbo, I don't think you can go wrong with the smiths on this site, with that said I'll give you a little advice on the selection of things for your build.
When I get a call at the shop about a new build for a first timer, They usually have a cartrige in mind. But their choice of cartrige may not necessarilly be a good choice for their shooting needs. Because uncle bob has a 300 mag or a xyz firedragon dosen't mean it's for them.
First question, Do you want a muzzle brake
? if you say yes it opens up some cartrige selections.
Second question, Do you handload? if you say yes it opens up more cartrige selections. You will find in long range consistant ammo is key to accuracy at LR, most factory fodder dosen't cut it. If your answer is no, match or specialty ammo is the way to go.
Third question, how much weight are you willing to carry? This may have some effect on the possible answer to question one as well.
If your looking for a 5-600 yard deer gun, you can pick from many different short or long action, non-magnum cartriges, with-out needing a brake, and weight could range from 7-10 lbs for a comfortable carry gun with some reach. 6.5's fill this nicely. Specialty ammo available in 6.5x284. Don't overlook a 308, quality match ammo is available and many deer have fallen out to 600, and the bonus is 4-5000 rounds of barrel life.
If your after a 600-1000 yard rifle that is capable of deer and an ocassional elk you would be in about the 7mm - 30 cal world, some type of magnum. A muzzle brake may be neccesary on the rifle depending on cartrige, and gun weight.
A 9 lb 7mm rem mag could be a good carry rifle with-out a brake. There are very good specialty ammo companys making loads for this cartrige.(BOTW offers a berger168 grain load (very consistant and accurate))
A 9lb 300 rum would most likly need a brake, but a 300wsm at 10-11 pounds could do with-out. Many options in this range of cartirges.
For the needs you stated in your first post, I'm not going to get into the 338s
As far as actions, I have no problem with the older remy actions, one of my favorite rifles, I built on a used adl I picked up for $275, add $150 for BP and a tunned factory trigger $40 and your at less than half of any custom. That 300 Rum is on my site in the gallery with groups shot at 100 and 300. A ligit sub 1/2 moa gun day in and day out.
The newer remingtons have bolt camming issues (as in lack of) and the price for new remmy's has also gone up. so if that is your only option I would look at a custom Remmy clone. Surgeon or Stiller.