It seems to me that when a class needs to contain data types that would preferably be forward declared in the header file, it would be most beneficial to just forward declare one structure to hold all of its member variables.

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Does this seem like sound design? I'm curious to know why it is not a more common practice.

It's actually quite common. It even has a bunch of names: the "pimpl idiom", the "compiler firewall idiom", the "cheshire cat idiom" and the "bridge pattern".

Keep in mind, however, that just because this pattern has some nice properties, it doesn't mean that this level of abstraction is necessarily appropriate or even desirable all the time. You need to consider how the class is to be used, its scope of responsibility, how stable its api is, and other things. For example, this wouldn't be appropriate for a 3D vector or matrix class, but might be appropriate for, say, a 3D renderer.