Matt, I'm sure some burl-wood would be strong enough for handles. I've used quite a few bits of different burls for marking-gauge stocks without any (known) problems, but I wouldn't trust most burl woods for the beams. If you can orientate the grain for maximum strength through the critical parts (especially the 'bridge' between the grip & the cheeks), you'll improve your chances, I think. However, I'd have to say it's dicey to use other than wood with a pretty predictable grain orientation. But what the heck, there is only one way to find out for sure.
I once made a small bowsaw frame from some very pretty Red-box burlish stuff (it was off a giant root-burl dug out by a Main Roads machine). Looked very spiffy, but even though I left a bit more wood where it mattered, it busted after very little use. Bowsaw frames are subjected to more severe forces than a backsaw handle, of course, but the experience did leave me with a lasting scar. I broke out a while back and used some Budgeroo 'root burl' for a handle, though I took great care to get as much long-grain as possible through the fragile zones. So far, it's holding up well, but I don't use that particular saw a lot, having a saw with identical specs, but a much more ordinary-looking handle to do the 'dirty work'...
Now, before I start sounding like I know all about it, I don't - shaping grips is something I learn a little bit more about every time I do it. I might be at the journeyman stage, but barely, and of course, personal preferences are paramount, so you can only talk generalities. Feel free to add your own observations/dissent! :U
I've discovered a bit of a 'rule' with handle thickness (or should I say re-discovered, because when I take a critical look at old handles from the 'golden era', I reckon the makers were well aware of it). There's a strong relationship between handle thickness and the width of the grip. If you check a bunch of old saw handles, you'll find the thickness varies a bit, but the more 'comfortable' ones tend to fall somewhere between 21 & 23mm thick, i.e. a tad less to a tad more than 7/8" in old money. A few are barely 20mm and that feels a bit skimpy to my hand, unless it's compensated by a slight increase in width. This is a 'trick' that can make a skinny handle feel fatter. It doesn't take much, even a mm is enough to make a big difference to how much it 'fills' your hand, and 2mm is usually quite enough - more than that & it changes the proportions too obviously, so there's not a big range to work in. The feeling of 'fullness' is also very dependent on how much you shape the edges of the grip. I like a more oval shape, because I reckon that fits the natural shape of your closed palm, but some prefer a more simple round-over look. Other subtle changes can make a difference in the way a grip feels, one is to vary the shape, from more roundish at the bottom, to more 'oval' up at the thumb/index-finger level, which I also think reflects anatomy better. However, the thickness/width ratio of the grip is definitely one of the major things to 'get right'.
It's worth mucking about with until you get what you think is perfect. I am quite convinced that if a saw feels just right in your hand, it will cut better & straighter for you.... :;
Cheers,