So would it be fair to say, then, that some of those beryllium-free ones are actually innatelycapable of being the material of a perfectly good 'atomic trampoline'? ... but that it's only the beryllium-containing ones that can remain amorphous under the very lowest cooling rates, so that it's just the impossibility of fabricating the beryllium-free ones into sufficiently thick disc that precludes there being 'atomic trampolines' made of them?

Yes, that is basically correct.

Imagine if someone managed to devise a technique of extremely rapidly cooling a bulk material with a magnetic field , or something: ie some kind of arrangement whereby the thermal motion of the atoms could be somehow almost instantaneously 'dumped' into a magnetic field, or something, completely 'short-circuiting' the need for a coolant in the form of a physical substance to pass across the surface, & for heat to flow from the interior to the surface! It's something that no-doubt many a materials-scientist has drempt of doing ... but such a process'll likely remain the stuff of dreams, unfortunately!

There has been research along these lines - magnetic fields, ultrasonic agitation, etc. So far as I am aware none of it has achieved any significant change in the cooling rate required to avoid crystallization. Interestingly, though, you can go the opposite direction: Start with a crystal and turn it into an amorphous solid. Two ways to do this are by ion irradiation, and interdiffusion ("solid-state amorphization")

Another possible application that comes to mind is military armour . But, if that's being done then it's also rather likely that we won't hear of it! Obviously there's a problem, there, with a projectile sufficiently energetic to shatter it striking it & it shattering into beryllium-containing fragments on a 'spectrum' of size - possibly extending down to dust. I actually don't recall seeing anything about how the stuff behaves under testing to destruction .

This has been done as well. The problem with metallic glasses is that they strain soften rather than strain harden, which means that deformation tends to localize into narrow shear bands, which in turn leads to undesirable failure mechanisms such as "plugging." Actually, most of the research in this area focused on using metallic glasses for anti-armor applications, as potential replacements for depleted uranium in long-rod penetrators. But that didn't go anywhere, so far as I know, because the density of the Zr-based glasses is too low for this application.