I understand the differences between the 700 and 10 actions.
Is there any reason to go with one brand over the other specifically as they relate to the barrel nut? Not triggers, stocks, etc.
My plan as of now is to buy the cheapest version of either when I find a good deal and put a prefit CBI barrel and a BC M40 stock on it.
My only goal is a rifle as accurate as possible for as cheap as possible. Not trying to break records just have fun shooting tiny groups.
I don't know if it matters but the caliber at this point is likely going to be a 6BR.
If you just want the cheapest I think Savage will be the way to go. Also, without doing additional work (trying, lapping, et ceters) I think Savage might be a little more accurate.
I think anytime you post something like this a lot of guys are going to tell you to get the action they prefer. In this scenario, based on the criteria you've listed, I think Savage is likely the better choice. I personally would prefer the Remington. I don't like Savage's bolt lift and I think most of their magazine options have feed issues, but I think you will likely find them to be more affordable and likely more accurate without doing work on them.