You can have bones that are parented but don't have one of their vertices directly attached to one of the parent's vertices. It's really just a hierarchy of transformation (child's overall transform = its transform relative to parent + parent's transform). Put differently, the "attached at the joints" notion is often approximated just because that's how human skeletons work, but it's by no means required. And "bones" in this sense don't always even correspond to actual skeletal bones; they're really just a tool for moving around groups of vertices without having to mess with hundreds or thousands of them individually.

That said - and I haven't done this myself, I'm just speculating - if you're using motion capture data directly maybe it makes sense to ditch the hierarchy altogether and have each bone be independent? Though, that could be problematic if you wanted to blend some "artificial" animations together with the recorded ones. I'm not sure what the standard practice is.

Raw motion capture data almost always requires some editing after it has been captured. This is done by the animator, and they can’t work without their rigs. Rigging is an entire craft... tried a enough to know that I suck at it. Rigging quadrupeds is a god level skill.