When you say "shave" you mean pressing the edge against your skin? Because when I try to take hair off with one of my knives, I typically don't touch my skin with the edge because I'm afraid it might just shave the skin off too

.
If I have a hard time getting my hair to pop off with an edge finished with my slotted paper wheel, I can't really imagine that something finished with 120 grit would do anything.
You can get a shaving sharp edge from a courser grit stone, if you do it correctly. One tutorial I watched shows the guy go from a rough stone with a knife that won't cut paper, to shaving his arm hair on just the medium/rough stone in a few minutes.
I can go from a 120 grit stone and jump the hair off the arm.
If I hit a few passes on a strop from a lower grit stone, then it is even sharper.
Of course, I don't leave my edges at 120 grit, but I could, depending on what I was cutting.
I do most my edge maintenance with a strop.
Nothing high tech or fancy.
I have a home made strop with 4 sides.
I use polishing compound from the hard ware store. The green bar (with a red #6 on it).
super cheap and effective.
Couple of common mistakes with a strop are
Pushing too hard. Too much pressure, and all you are really doing is dulling and polishing the edge. The leather has enough give that it will wrap back over the edge and actually dull the edge.
If you are doing 40+ passes per side, you may be getting sloppy with technique.
If the knife is reasonably sharp already.
Second big mistake with strops, is simply wrong angle.
Sounds like you may have figured it out with the lighter touch and fewer passes.
You can polish all day on the strop without dulling it, but it won't be getting any sharper either.
One of the sharpest knives I have is one I took from a worn 80 grit belt right to my strop (the belt was worn).
It was a shallow angle convex edge, fully blended bevel with a flat grind in 5160 steel.
Very scary sharp. (of course, 5160, with a basic back yard heat treat and temper is very easy steel to get starp. It was polished in no time. After a little while of use and stropping, it looks like I finished the edge on 1000 grit sandpaper before stropping).
I hear a lot of people refer to their edges as "scary" sharp, and when I test them, they rarely are.
One knife I got had a beautiful polished convex edge. Nice blend to the primary flat grind. The previous owner said it was "scary sharp".
It was dull. Would not shave hair with significant pressure. A sharp knife will shave the hair without actually touching the skin. (Tree topping). You don't even need to touch the edge to the skin. That is when you start to get scary sharp.
With most steels, this edge won't last super long. But I find that steel and angle being equal, a nice polished edge will stay "usefully" sharp longer than a rougher toothy edge (this depends on what you are cutting also).
Glad to hear you are giving your strops another chance.
You can get sharp with so many different methods. Just use what works for you.
I find that when you get good with one method, it may actually help with other methods.
Sandpaper and mousepad, and strop made me much better with a flat diamond stone as well.
With all the quality stuff you have for the strop, you should be able to get laser edges.
I think, including the sandpaper, sheet of glass, and home made strop and cheapo polishing compound, I may have 15 to 20 bucks invested in my kit. People, even "knife" people are scared of my edges. My dad and brother won't "test" my edges with their thumbs because they are that sharp, and they are knife people, who carry and use knives every day.
If you were in my area, I would have a get together and just check your technique.
At some point, with a dull edge, you may have to drop back down to either a stone, or stick, or sandpaper, then go back to the strop.
You can take a dull edge back to sharp with just stropping (very dull edge), but it takes forever, and if you have any edge damage you basically have to go back to the rougher stuff to fix it first.