Re: .223 bolt action on a budget
While I gladly 2nd (3rd/4th/5th) the Tikka recommendation, the only problem I see with it is your total budget of only $600. The Tikka (either the Varmint or the Scout) will be roughly $200 over your buget at around $800-850. Now, you obviously get what you pay for with the Tikka including arguably superior components (especially the stock which is a real POS on the Remy) and an overall superior product, but if the money isn't available, there is nothing per se wrong with the SPS Tactical .223 other than the stock and the slower twist barrel which limits your bullets selection somewhat (certainly to 68-69gr and less). Since I prefer the 75-77gr pills, it is a deal-breaker for me, but for shorter range work, you will be fine with the slower twist running 69gr and lighter pills unless you just WANT to run the heavier ones.
As for the Remington LTR (or a PSS/P), I don't know that you will find one (even a good used one) for much less than a NIB Tikka. While the stock on the LTR/PSS/P models (HS Precision) is a big improvement over the SPS stock, you are still stuck with a slower twist barrel.
Again, it all comes down to what specifically you want/need from the rifle and how much (if you want to go over your current budget) you are willing to spend.