I think the strength comes from being forced to defend your positions. Any paper worth its salt anticipates and defuses most likely objections. It really does force away any areas where one might have hand-waved.

I think (as someone else mentioned too) teaching is generally good because one never knows where and when a challenge may present itself, but I'd suspect a paper gives deeper results, whereas teaching gives broader results, and both give more than study with neither.

If you think writing prose solidifies your understanding of a subject, try formalizing its findings as knowledge .. in code.

Most specifications can be turned into a database for some form of Prolog, play with it. At the very least, you might get yourself a type or model checker, or you might get lucky and actually get an expert system out of it.

This is what I've found about programming as well. Computers are unforgiving of any gaps in your understanding, so explaining a process to a computer improves both you and the computer.

Of course, not all ideas can be so neatly expressed in code. Constructing essays and explanations for people is similar, though it's easier to finesse gaps with rhetorical tricks.