Yup, these guys know their stuff.
Bores on rifles chambered for 303Brit can vary from as tight as .309 to .318. Those are extremes though and rarely found.
Most are not bad and range between .310 and .314. There was a time, when CIL and Dominion made up bullets for the 303 in 5 different diameters. They knew what they were up against back then and wanted their bullets to shoot as well as the rifles available were capable of.
Lee Enfields of every stripe will shoot very well when they are well maintained, bedding kept tight and proper diameter bullets over adequate charges of powder are combined. Any one of these is out of spec and you have accuracy issues.
I have some US made 303Brit ammunition, dated 1943. I pulled a couple apart to see how they were made. They had a ball powder that may have been BLC and 175 grain flat based .308 diameter bullets and were Boxer primed. They were OK in some rifles at best except for one in particular, which I still have, bought it in the grease for $15 from a barrel in the local Hudson's Bay Store. This rifle is the exception to the rule. It's a 1944 dated Long Branch with a 2 groove barrel. It will shoot just about any .308 bullet, other than the very heavy 200 grn plus into 2inches consistently. Even the 135 grn, hand swaged bullets for Hunter Bench Rest. It is the exception again, with a .309 bore.
Some of the excellent Pakistani No4 MkII rifles that were common a couple of years ago had .310 bores and were OK with .308 bullets but preferred .311. Mine will not shoot .312 diameter bullets well at all.
OP, your No5 can be a dream or a jack handle. So many variables that may have nothing at all to do with the cartridges it is being fed that may effect accuracy.
The 303 is a great cartridge. During the last decade, it has had a new awakening amongst collectors and shooters. Finding decent rifles chambered for it can be a challenge though.
Good luck with it. If you are having accuracy problems, there are several Nutters in the Edmonton area that really seem to know their stuff when it comes to 303 chambered rifles. Query them on the site for help. Rebedding the Lee Enfield series of rifles can be tricky but it's pretty straight forward. The MILSURP KNOWLEDGE site has detailed instruction on what to look for and how to detect/fix worn bedding.
Not difficult, just tedious. Once it's finished though, the results are well worth the effort. Even with a worn barrel.