How can I justify doing a second Master's in Mathematics? I'd like to apply for a PhD in Mathematics/Applied Mathematics in Europe. I have a BSc in Math and a taught 1-year MSc in Applied Math (without thesis) both from top US universities. I have taken advanced courses during my undergrad, I do have research experience, but not in Mathematics (it was in Data Science, mostly involving computational programming) and have no independent research experience (all of it was directed research).

Now I'm looking for ways to make myself a more competitive applicant, for which I asked for suggestions in this questions. But here, I wanted to ask specifically about doing a second Master's in Mathematics. 

I have seen this question, but this doesn't apply to my case, since I may actually pursue a Master's in a more general field (Mathematics instead of Applied Math). I've also seen this one and this one, but these two don't apply to my case either, since I'm actually open to getting my second Master's from a "lower ranked" university (especially since I'd most probably need a scholarship, which would be hard to get from a university at a level of reputation comparable to that of my undergrad and MSc institutions). 

Do you think a second Master's, most probably in pure Math (not Applied Math), and most definitely with a substantial thesis component, would help me to get into a competitive PhD program (in Math or Applied Math) in Europe? If yes, how do you think I can justify that in my SOP? My personal justification is the thesis, and the fact that most of my MSc courses were in Differential Equations and Numerics (and very applied stuff in general), and I definitely miss graduate-level knowledge in areas such as Algebra, Analysis (proof-based side of it), and Geometry. But would that be acceptable for the committee and would I have any chance for a scholarship with that justification?!

Edit: As for the specific countries in Europe I'm looking for, my top choices are Germany, Switzerland, UK, and France, but I'm open to other options if I find a good opportunity.
(Note that my perspective is US.) I think your first priority should be applying directly for the doctorate. You can simultaneously apply for an MS, of course, but I suspect that you are underestimating your chances for doctoral admission. 

Yes, candidates are expected to have a pretty broad education in math on entry. Yours seems pretty broad if a bit non standard and you are missing a few things, but you won't know until you apply. 

Doctoral studies are almost always about specialization, however. If you intend to work in an area in which you are already prepared, and say so in your application, then the missing bits might be overlooked. Not necessarily so, but you won't know unless you get into the process. 

A second master's will broaden your education a bit. It may make a difference. Or not. Many places (US, at least) will treat the Bachelor's degree as enough. UK and EU may be more strict.