This isn't really a question of whether do can substitute for an -ing form but of reading from context.
The best interpretation for does is "mentions the weakness". This is because the context is stealing someone's thunder, an idiom which means "to take someone else's idea" or "to destroy the effect of what someone else does by doing it first."
So the advice to lawyers is to steal the opponent's thunder by mentioning the weakness first: if they mention the weakness first, it won't have much effect if the opponent brings it up later. It doesn't make sense to think of the opponent as "stealing someone's thunder", because they're not taking your idea, you're taking theirs.
To address the question more generally, do can substitute for an -ing form, but usually not directly or without changing the structure a bit. For example, you could say
I am taking this candy before the children do
but you wouldn't literally write
I am taking this candy before the children taking it
I am taking this candy before the children are taking it
You would have to write
I am taking this candy before the children take it