il take a look at flex cut, sounds promising.
In general sharpening by stone is the sharpest way and stropping overtime will eventually dull the blade and then it goes back on the stone and repeats the process?
This is my take.
Sharpening by stone makes the flatest possible cutting plane. When you strop on a surface with give, the edge is going to take on the shape of the arc defined by the amount of give (how soft the strop surface is and how much force is being applied).
The strop surface is somewhat conformable and it wants to spring back to its original volume as the edge passes over. So in addition to any overall convexity, you can have even greater rounding right at the cutting edge until the shape of the edge comes into equilibrium with the arc defined by the strop surface.
If you're only giving it a few light passes off the stone this isn't really an issue, if you're doing it constantly, over time it can make your edge way more broad than is good for effective cutting. Using a harder backing can help this out considerably. Since it won't deflect as much, one can apply a modest amount of additional force, and/or it will be more tolerant of variations in applied force from repetitive use - it will come into equilibrium with a much more shallow arc.
This also allows the compound to do more restorative work. Very handy, as stropping generally doesn't cause anywhere near the level of burring a hard stone will, so less time spent on QC and less need for fine angle control as the slight bit of deflection (hopefully no more than a degree or so) increases the margin of error you need to hold the edge at and still have the compound working effectively at the apex.
Also, most stropping tends to reduce all the fine irregularities along the cutting edge left over from the stone work - not necessarily a benefit for EDU, but for slicing leather you're probably better off with a thin ground edge that is more refined/uniform anyway.
Flexcut, being formulated for wood carvers, does a good job in terms of prolonged maintenance as it has a small percentage of larger grit for "grinding" a fresh cutting edge, and a large percentage of finer grit for refining it - all in one step. This is the same strategy I use in my custom honing compound included with the kit I make. Did a lot of trials and for general maintenance use, the multi grit formulations outperformed the single grit ones every time.
If you do simply apply the compound to a sheet of paper and wrap around the stone, I highly recommend using the coarse side. It allows the paper to deflect into the low spots on the stone and it won't push back up so hard as the edge passes over. The spots that do ride on the high points in the stone will permanently compress the paper a bit more than they would over a flat surface, and this action reduces the tendency to expand back up at the contact points as the edge passes over (another trait I adopted when making my Washboard). Just brush off the stone - a few swipes with your hand will do - before using, as any loose grit trapped under the paper can deform the cutting edge with surprisingly little appled force.
I prefer to use paper over a stone or one of my Washboards, because once you start to do more stock removal with a compound, it will load up much more rapidly than it will when used for a light final polish. Instead of having to clean it off all the time, just toss the paper and start fresh with a new sheet. If you opt for a leather strop make sure it is well cased and very hard. Balsa is another common choice.