I usually work on the nut behind the trigger, then move on to others screws, then glass
Not to slam the OP, I don't know him or his background. But I've been at the range a whole bunch of times and heard how bad a rifle/pistol performed, only to change shooters and watch it shoot very well. I used to work with a guy that that went through 5-7 expensive handguns every year, because they all shot low and to the left. And by golly it was always a bad gun, never a bad shooter.
"Accuracy" to me is repeat-ability. A rifle can't be accurate, if it only does it once or twice. Hitting your target once, missing every other time, to me that's luck. It's by definition, inaccurate.
.223 is such a mild round the barrel would have to be OMG levels of garbage, maybe made out of tinfoil and expanding foam, for 3 rounds to generate enough heat to be an issue. It's not the first thing I would suspect if it were my gun, good chance it would be the last. But I'd still run a dollar down the barrel as a start.
If it did turn out to be a bad barrel, it would probably join my wall of fame. It could live forever right next to my favorite scope that I'll never part with.
I've owned that scope for about 20 years, 18 or so in it's current form