Can research breadth be a downside when applying for doctoral positions?
You could say something like:

"I've done a pretty wide range of different things, trying to find out where my true passion lies. Getting multiple masters degrees shows that I'm not a dilettante, I get stuff done. And it's been an enriching experience, letting me see the connections between different fields. But after long consideration I've decided pure math is what I want to settle down on."

And it has the virtue of being true (from what I read in your question).



Consider what a professor might think, while reviewing two candidates, one pure math straight up, and you who's tried several different things and finished them.


The pure math student might have a lead on you in pure math.
You've demonstrated persistence and an ability to get things done. Likely, you will also get a PhD done.
You've shown that you're mentally flexible enough to master multiple different subject fields.
The math student might at some point decide he's not seen enough of the rest of the world and worry if staying cocooned in pure math is really what he wants. You on the other hand have tasted the various forbidden fruits and decided this is really the one you want.


Just on the surface like that, you look like a candidate worth interviewing.