Slugging the barrel is never a bad idea. It's better to know than to not know.
I shot a few .308 bullet reloads from mine, and didn't notice any problem with accuracy. I didn't chrono them, so I don't know if there was any loss of velocity. (doubt it)
Mine slugged between .312 and .313. When I'm loading cast bullets, I won't try any that are smaller than .313... .314-.315 is better.
In my experience, jacketed bullets are not that finicky...
So what is the best way to proceed? Should I slug each barrel and then load based on that? As I understand it, the process is to ram a lead sinker through the barrel and then measure its width with calipers.
The process you quoted is pretty accurate. As long as the lead slug is a few thousandths bigger than you expect the bore to be, you'll be golden. I recommend doing it at least twice, to help eliminate errors.