There is nothing magical.. nor difficult about reaching out to 1000yds with a 7RM. In fact, all the hard work is done at 350yds.... the rest is just adding more elevation and leaning into the wind a bit more. Really... it is that simple.
Every factory Rem 700 trigger I have seen can be adjusted for pull weight. If you get it down to something you can break cleanly, good enough.
Assume you reload... and if LR shooting is your new hobby, then reloading will be critical. Actually more important then a new rifle.
Load a 162 to 175gr match bullet, load to a moderate (mid load) using printed data, get a load that is consistent... CONSISTENT is the key. MOA or better for 4 rds... 5 rds if a heavier barrel. If the groups open quickly once you shoot 3rds, then move to a new match quality pipe asap.
Otherwise, from your 350yds zero, dial up another (whatever your ballistics programs suggests)... pull the trigger, that's it. Heck, if you have a hunting load that you feel confident in, just dial up and send it.... 1000yds isn't going to tax any bullet you are likely to shoot from a 7RM at typical speeds.
Order of things to do with the current rifle
1 - properly bed the action/stock
2 - lower trigger pull weight... should be able to just turn an allen bolt if a newer trigger
3 - load a match bullet to a consistent moa or better
Go out and have alot of fun.... and while you are actually shooting that far, you will figure what you actually need/want. Recoil is a big reason I got out of boomers once we had the bullets to make smaller cals rock. My goto is a 22 CM running 95gr Sierra MKs at very moderate speeds... out to 1500yds, it is nipping at the heals of a 408CT.
Todays darling is the 6.5CM... yesterdays is the 308Win... both will make 1500yds easy work with todays bullets. at 1000yds, both are more then capable of sub 1/2 MOA accuracy all day long.. if you can drive it.
optics matter but high mag is rarely what you want for field shooting. if you are shooting paper targets which are scored, like F class, then a higher mag scope is helpful. for field shooting, high mag is NOT. Clarity and resolution and wide field of view are far more important.
Athlon has a number of amazing scopes that will let you shoot with 100% confidence and success ranging from not much money to quite reasonable. Most of my LR and ELR shooting is now done around 16X... on occasion 24X and even down to 12X. Glass quality is now good enough to make this type of shooting viable without high mag. I would lean you immediately to the scopes with a 34mm tube... every single one from Athlon will offer you all the performance you need to make hits way the heck out there.
If you are going to build up a rifle for 1000yds+ shooting, I would suggest you reconsider your chamber choice. I enjoyed a number of 7RMs in years past and they shot really well but with todays bullets, you just don't need the expense and consumption of components and barrels.
The most important part of your build is the barrel. There are PLENTY of excellent choices... and a quality match barrel is what allows you to load to itty bitty groups. Modern actions that are built well and can shoot well are pretty much the norm so whether you go factory or custom, this is not where you will gain accuracy.
Chassis/stocks is like clothing... wear what you like and feel comfy in. Again, plenty of excellent options that have the features to work.. and cost ranges from low to high.
Things that can limit your action choice will be things like trigger pull weight. If you want a specific type of stock/chassis with limited inletting options. Or just personal tastes... at the end, you have to like the rifle you bring to the range/field.
Reaching out to 1000yds is like trying to find a car that can drive 120km on the highway... lots of options and plenty of ways to get there.
Things get more complicated if you wanted to reach out to 1 mile and beyond... but the K is a chip shot with modern gear.
In case you aren't familiar with MOA as it relates to target size, MOA at 1000yds is a milk jug.... if you have a set up that will CONSISTENTLY shoot MOA or better at 350yds, you can hit that milk jug at 1000yds once you know your dope.... with the same level of impacts as a MOA target at 350yds.
Simple....
Jerry