The Springfield O3 model/variations have a reputation for accuracy. In many cases, it is well deserved.
Like most military rifles, if all is well as far as bore erosion, bedding, diameter of bore etc the rifles will shoot within specified parameters with issue grade ammo.
Take one of those rifles and feed it with ammo it likes and it can be a virtual tack driver as long as the operator has the skill set to make it perform.
I have had some spectacular 03s and some real dogs. I remember a couple of pallets of them that came in from some off shore nation that either got them as lend lease or picked them up from a broker for their military. Whoever they came from, respected them and took very good care of them. All were worn but well maintained with original barrels and relatively sharp bores. All had been counter bored. This could be for several reasons but usually it means the rifles have had their muzzles and last bit of bore rusted or the troops cleaning them with pull through cleaning kits weren't careful to use a muzzle guide and dragged the ropes against the muzzle in a rush to get the job done.
I was looking for a decent one as a shooter and had the opportunity to slug the bores of the selected rifles I had set aside before sending them up to the store for the racks. Only one of the ten or so rifles had a bore that measured reasonably well. IIRC, it was still about a thousandth large. It was well used so that was no surprise. I sent all of them upstairs and declined taking one. The next batch was almost as bad other than one gem that somehow made it to disposal. It was VG on the outside and had a perfect bore dimension. I bought it.
I took it, along with a couple of others that looked good to the range (100yds) to see if they would shoot. The "gem" was the worst performer of all of them. None of them would shoot better than about 4 inches. The rifle I had purchased had a couple of bedding issues the were easily corrected and later performed quite well with surplus ammo, better with selected handloads.
What I am saying, is the 03 rifles, no matter which variation, perform about as well as any other military rifles. A lot of the "most accurate rifle in the world" is just hype to build up the confidence of the troops using it as well as the general publics impression of the arm being issued to its protectors.
Do a complete rebuild on just about any milsurp, do it right and they will shoot extremely well if you feed it good ammo and do your part.
OP, that looks like a fun project. My only advice to you is to do a cost evaluation before hand. If you plan to keep it in full military configuration be prepared to put out as much money on it or more than a sporting rifle built up on a modern receiver. If that isn't a deterrent go for it.
I have a project rifle very similar to the rifle of the Fort Worth poster sitting with all of its parts waiting for final assembly. I have been trying to decide which trigger guard set to use. I have one of both in new condition. Many folks with the No4 MkI* Long Branch rifles change out the stamped trigger guard assemblies for the stiffer milled assemblies because they feel they are stiffer. Oh well, at least I can swap out one for the other, depending on which does the best.
Before the weenies start, this rifle was rescued from a fellow that was going to turn it in to an amnesty several years ago. It is a Remington 03A3 with a perfect bore and sits in a very nice Bishop replacement sporting stock. It was drilled and tapped for a scope and the bolt was ground to clear a scope. I have a new old stock surplus, type C stock and new metal to put onto this rifle. Can't decide if I want to scope it our set it up in a similar fashion to the Ft Worth example. Will just have to see how the wind blows.