I wanted to add a couple things as I've done some experimenting on barrels past their peak, and with what I've learned scoping & testing them there are things I carried to brand new barrel maintenance.
For one, you need to be leery of terms generalized and research to understanding.
-In the Lilja video, he last shows a brand new bore and describes it as 'mirror finish'. This finish is NOT ideal, and I suspect nothing more than merchandising on their part. Only the lack of other problems holds as good.
-In the Hawkeye video, notice in the last example a 'tack driver'. It is not a polished(mirror) finish. It has a CORRECT surface profile for reduced fouling & lasting performance(grouping).
A bore that shoots really good is not perfect as possible,, it is just right. A balance.
And this is why you need to know what you're doing if you're going to do anything with it.
I use Tubbs FF. It has proven(to me) to work well for new barrel break-in as it lays a lap similar to the best hand lapped bores. I shoot a couple of their TMS bullets every 200-300rnds to dress up lands(they start uneven over time). I haven't seen, or read of Tubb's firelapping hurting any bore, or reducing any barrel life. The truth is this system extends useful life of bores, when used as it should be.
A lot of folks are afraid of this, I was at first, and watched it close with a Hawkeye. No problems.
Also their are 2 different JB compounds out there, both claiming to polish bores. Be careful to get the right one, as one of them actually does polish bores and will instantly cause copper fouling to forever increase.