The only modern bolt gun I can think of that comes with military sights is a Steyr Scout.
Bolt-action sporters have never been common with sights like a No 4 Enfield or an AR.
Those sights are made with a different purpose in mind: hitting a man-sized target somewhere, at a wide range of distances. They're not meant for bullseye shooting or ethical big game hunting at a distance.
Peep sights are a popular aftermarket item, though.
Now iron sights like a Mosin or a No1 Mk3? Those are common on most bolt guns, in some configuration or other.
A Ruger 77 International, Remington Model 7 MS (Custom Shop), a full-stock CZ, etc. are some of the closest modern rifles to an old military rifle in appearance and feel, without the ungodly price tags of some other European guns.
EDIT: Oh wait. I just saw the price of the Remington. I didn't know they were THAT much. Scratch the Model 7 MS.