Gator's solution is a long-used and very effective method to remove lead fouling. If you can find it, brass or bronze chore boy/wool works just as well. This is the lead removal recommended for years by Veral Smith in his book, Jacketed Performance with Cast Bullets. In my opinion, removing lead this way is more effective than firing a jacketed bullet.
Now, I must admit, in my many years of black powder shooting, I have never measured the "flame temperature" of my loads, but do you seriously think the interior barrel walls reach over 650 F after 6 BP rounds and stays hot long enough to soften lead deposits? And since the OP was asking about jacketed bullets, shotgun barrels really don't enter the discussion. Ten rounds of 38 grains of FFFg through my 44-40 rifle barrel get it pretty warm, but I can still hold it with my bare hand.
Are you now suggesting that it is not possible to lead the barrel of a gun when loaded with black powder? That is simply not the case.
You are certainly entitled to your opinions, but I was interested in the facts to support them.
An area where we agree is that by using the right bullet diameter and material, lubrication, bore condition and pressure, lead fouling will not be an issue.