I've evolved from the scabbard issue
I used scabards from 7 years ago, up to 2 years ago as I was exhaustively testing the brands and styles of hand saws availavble to arborists. I smile fondly in reading this thread, seeing the innovations.... I really enjoyed those periods of modifying something to work better than it was designed as many of us do.
I still am constantly modifying gear to be more efficient and effective, but as far as the handsaw thing, I don't use a scabbard anymore. Same complaints as what I hear from you (hanging up on rope, branches, etc) The GREAT thing about a scabbarded saw it that it's right there - Quickdraw McGraw. My Silky Zubat was my favorite as I found the slightly larger Ibuki, well, a little too big for what I was doing. Also, both of these are curved blade saws.
I moved on to the straight blade Masaru, the same length and tooth style as the Zubat, and also in a scabbard. This excelled at doing face-cut wedges and back cuts for getting branches and small limbs to fell the direction you want them, and to snap off predictably. Still, the scabbard caused me the mild annoyance that I could see no way around (still, it's a rather minor annoyance. Couldn't re-scabbard the saw completely 'by feel' (without looking).
Then I entered into the Silky GOMBOY family. These are four folding saws, each a little bigger than the previous, and all have the awesome Zubat-style teeth. The biggest is 330 mm and this is currently my favorite folding saw. Early on, I just threw it down the front of my shirt and did the climb. Now what I've done is put two wraps of 14 gauge stainless steel wire through the hole at the end of the handle to create a stiff, one inch loop. This then can be clipped to the rear of my saddle. I can 'on' and 'off' it one handed, unfolding the blade is almost instantaneous and folding it up and clipping it back on takes maybe a couple seconds longer than a scabbard. But, I can do it without looking.
This system has become such a smooth and efficient mainstay for me, that I gave up using a top-handle chainsaw about 6 months ago in leiu of one with a bit more weight and a lot more power (Husky 346XP). With this setup, I have never been more effective up in a tree. In limbing, like John Paul, I like to make my face cuts, start my back cut, shut off the saw and finish the back cut with the hand saw. Much better control. Much safer, much more boiled down. I've since mounted a scabbard on the chipper and it's right there to Quickdraw McGraw the forked limbs that aren't like likely to collapse going in. A quick relief cut at the fork, and continue the feed. It's either that, or wrestle with the limb or fire up a chainsaw. I thoroughly love my Silky Saws.