Give yourself a big enough target, and meter your expectations appropriately, and any 6.5 creedmoor which prints 1-1.5” groups at 100yrds will let you casually plink at 1,000yrds on big targets.
More expensive bolt action rifles will be a lot better for the task.
This is a good and realistic point. Not everyone who wants to try shooting at long range is going to be jumping head first into a gun game like PRS and shooting tons of matches around the country, expectations and goals matter.
While there is truth to the statement that if you get into the long range games you'll probably want to upgrade at some point, that doesn't make the purchase of a cheaper starter rifle a waste. I started with a Tikka CTR .308 and an SWFA SS fixed 12 mil/mil. I had fun shooting a few matches and working on my techniques out to 800 yds at the local range. I could have bought pretty much any rig out there, but there's no way I would have dropped $7k right out of the gate, the $1,200 price of that setup was stomachable for a newb looking to try something different. No one was offering to just lend me their long range rig for a few months to let my try it out, so the choice was either the starter rifle or a different hobby.
I still shoot a starter rifle, an RPR, but have a much nicer scope now, the fixed 12x and the SWFA SS 3-15 that was on the RPR first are still chugging along on ARs. I've won a few local matches and get out to my 1,000 yd range most weekends outside of hunting season. I'm sure I wouldn't be competitive at a national PRS match, but I can be competitive at my local matches and always enjoy my time shooting at the range and out on public land, those are my goals and I'm not sure a custom Impact rig would better help me meet them.
All of that said, the OP should definitely ditch the break action idea and pick up a decent rifle and starter scope instead.
Some good starter rifle options would be:
Ruger Precision Rifle
Tikka CTR
Bergara B-14 HMR
I'd go 6mm or 6.5mm Creedmoor, they are the easy button chamberings, good factory ammo available everywhere and they are much better than .308 at long range.
For cheaper scopes you either want a fixed power or a first focal plane so that your reticle remains useable at all magnifications. Reliability is key, if your scope won't hold zero, or doesn't have repeatable adjustments, it and any rifle it's attached to are useless. Reliability, glass quality and extra features all cost money, the only one you absolutely have to have is reliability.
This is where the SWFA SS series of scopes come in, the fixed power scopes are pretty much the bottom floor for reliable dialing scopes. The reticles are a bit simplistic, and the turrets are only 5 mil/rotation, but they work and if for some reason they ever cease to work, SWFA replaces them. For a little over twice the price, the SWFA FFP 3-15x42 gives you variable power with reliability, features are still limited and image quality is adequate. Above that you get into Burris XTR II, PST II, Athlon Chronos and Ares ETR, etc.
The point is, if you are interested in long range shooting, don't let rumminations on having to spend several thousand dollars out of the gate scare you off, you can have fun and start building your skills for a $1,500 initial investment or even less if you buy used or catch some deals.
