Why wouldn't you?! (Whether anybody else did or not)
1. Completely unnecessary.
2. Good chance of loosing small parts.
3. You either need a screwdriver with a slot in the middle, or you have to buy their special tool. As a matter of fact, that tool they are selling must be a relatively new item, I have never seen it before. The Brits call that a Spanner.
Seriously, I bought my Taylors (R&D) cylinder for my 1858 EuroArms Remmie a long time ago. Not sure exactly when, probably around 15 years ago. I never shoot it with anything other than 45 Schofield cartridges loaded with Black Powder.
The cartridges should seal the back of the cylinder, keeping BP fouling away from the firing pins. I don't see why the backplate (with the pins) should be routinely disassembled.
No, the cartridge case will not completely seal the chamber, some soot will find its way back to the area on the cap where the firing pins sit.
That having been said, I have never bothered to remove the firing pins for cleaning. All I do is give the back of the cap a good scrub with with a bronze bristle brush and my favorite water based Black Powder solvent. Give the firing pins a good squirt of canned air to dry them, and maybe a drop of BP compatible oil (Ballistol) to prevent corrosion.
I don't know if the new ones are any different, but mine has one shiny ferrule for the firing pins. Probably Stainless Steel. The idea is if you are only going to load five rounds (which I recommend), you leave the chamber under the shiny ferrule empty, so you know where the empty chamber is. That way when the cylinder is closed up you will know where the empty chamber is.
You can also use the 'viewing windows' on the sides of the chambers to see if there is a rim in there. They are not really viewing windows, they are clearance cuts because when the counterbores for the rims are cut, there would be a paper thin bit of steel left there, so that amount is cleared away.
One other thing. I strongly suggest not dry firing. There are no springs in the firing pins, just the pin and the ferrule. The pin is a loose fit in the ferrule. When you pop the back plate onto a loaded cylinder, the pins slide back, allowing the cap to seat all the way. Unless the design has changed, dry firing will jam the firing pin in the forward position and there will be much gnashing of teeth as you try to force it back so the cylinder can be capped. Also, over time, the rear face of the firing pins may tend to peen over from being struck a bazillion times by the hammer. That is the only time the firing pins need to be removed. So far, mine are still working fine.
P.S. I should add, I find it much easier to clean Black Powder fouling from the cylinder of a cartridge revolver than from a Cap & Ball revolver. The chambers are wide open and you can run a cleaning brush completely through, and scrub off any fouling in a jiffy with a bronze bore brush and your favorite water based BP solvent. No little nooks and crannies like there are in the multiple diameters of a C&B chamber, and you cannot run a bore brush completely through a C&B chamber like you can with a cartridge chamber. I always use a whole bunch of Q-Tips when cleaning a C&B cylinder.
P.P.S. Apologies to 45 Dragoon who has probably forgotten more about C&B revolvers than I will ever know.