For me Ball Centers are no more than a Stub Taper Shank, make them or buy Cheep Shanks for mounting Drill Chucks.  Center Drill with about a 3/8" dia. Drill and also have a corresponding Center in Both Ends of the Barrel.  Get some Bearing Balls to go between and your good to go.  Unless you're going to run High RPM grease in the Center Hole will be fine.  Just set up a Barrel but I'll post a picture later.
Congratulations on finding out what worked👍🏻. David

Somehow I kinda doubt Pope had ball bearings running in a cone.

Whatever you've cut in the barrel, is going to wear/deform to mate in some way to the hardened center in an offset tailstock. In theory... or maybe just my imagination... the center deforms the cone into a radiused ... 'something' ...

I'd have to ping a couple friends to know whether its a parabola, or a section of an ellipse, or ... ???? Hyperbola?

So ... my limited experience is cut a section for a steady rest and rough it in. True that section up after roughing with however light a cut you need to avoid chatter ... 'resonant frequency' ... play with RPM ... likely SLOW to get a decent cut. And watch the preload on the tailstock center as the barrel warms up and expands.

Once again my opinion 🤔. If your Lathe has a Chip Pan and you can set up for the Coolant Pump definitely! If you plan on Chambering, then a constant flood of Cutting Oil helps.  If you ever do a Stainless Steel Barrel then for sure you're going to need Cutting Oil.  Dipping a Brush is o.k. but I'm lazy.  The Spray Mist might be good on a Mill/Drill but I don't like breathing in the Mist.  David

No way I'd get that anywhere near a tablesaw.

I've used high rake polished inserts for Aluminum to turn 416. A lot lower pressure. Typical inserts for steel aren't as sharp, with the honing of the edge and buildup of the coatings.

I also have a sleeve to run in the steady rest, with 4 screws at the ends to dial it in. I've seen it need some tweaking after roughing in, the blank warping slightly.

That big chunk of lead.

I do tapered barrels an alternate way, I know, I know, I'm not a machinist.

But, I've never had to deal with chatter on my light Atlas lathes I used.

I don't set over the tailstock.  I chuck the barrel on centers, and turn it concentric.  If the breech end is concentric with the bore, hold it in a 3 jaw chuck to drive.  If it isn't, I indicate the barrel in using a range rod and/or a gauge pin in the bore, and turn it concentric.  That way, I can drive the barrel with the breech held in a 3 jaw chuck.    Then, I turn the barrel to the largest diameter, using a sharp V tipped HSS cutter.  Leaves a rough turned outside, like very fine threading.   

Next, I take a yardstick, and a sharpie and make a mark every inch along the barrel.  Calculate the taper of the barrel in thousandths per inch of barrel.  Say, it comes out .016" per inch.  Start at the muzzle end, and start with a .008" cut.  As the carriage moves, turn out the cross feed at a rate that will take out .008" per inch of travel.  You'll cut a taper for 1 inch, and then it won't cut.  Restart, and feed in .016".  Again, turn out the cross feed at .008" per inch.  This time you'll cut 2 inches along the barrel.  Now, remark the 1 inch marks you've cut off, so you can estimate the feed.  Repeat this, increasing the feed by .008" each time, until you have the entire taper cut.   

This is faster than the time it takes to set over the tailstock, check it, and reset it and check it when you're done.  You can do a taper amazingly accurate by this method.  At first, you're not very precise, but by the time you're repeated it 25 times along the barrel taper, you're pretty good and you'll come out with a uniform, smooth taper, ready to clean up with a file and start polishing.  For some reason, the changing feed on the cross screw seems to suppress the chatter and make the cut come out better.  Probably because non-linearity in the hand feed adjustment cancels out the harmonic chatter.

I then file it while turning at slow speed, holding the file at a 20 degree or so angle, and feeding towards the headstock.  Then, I take a couple 3 or 4 inch wide pieces of 1" boards, clamp them together, and drill a hole slightly larger than the largest part of the barrel, centered on the seam.  You now have a half round hole in the board to hold sandpaper to sand the barrel, again at slow speed and feeding toward the headstock while holding moderate pressure.

I have never had use for a ball center. I use a dead center for the head stock and an adjustable offset center for the tail stock. This is a tool I designed and have been using for over 20 years. I gave the idea to PTG and they started making them but it is vary hard to find them on their web page. This center is mounted in a double row ball bearing and will take all the thrust I can put on it. Anyone with a lathe and mill can make one.

[ftp]https://pacifictoolandgauge.com/lathe-jigs-kits/2019-ptg-adjustable-tailstock.html/ftp]