Age: 25 now, will be 27 when I hopefully enter law school.
LSAT: 180
Undergrad: Imperial College London, 80%, Mathematics BSc First Class Honours (LSAC translated this to "Superior"), with expository thesis. (Completed 2014)
Masters: Cambridge, Part III Mathematics, Distinction, with expository thesis. (Completed 2015)
PhD: Oxford, Pure Mathematics, two publications. (Completed 2018)
Work Experience: Currently a quantitative researcher at a very well known (amongst finance people at least) hedge fund. I will be working here for ~2 more years, running this gig so I can graduate from law school debt free. Internships in software engineering and quant trading.
Softs: Multiple marathons and triathlons including one ultramarathon and one (full) Ironman triathlon. Fluent in French. LRSM Piano Performance (very roughly the level of performance one is expected to have by the end of their 3rd year at a conservatoire).
Target schools: Honestly, I really want HYS

Concerns:
1. International student, non URM.
2. Focus on mathematics so far in my studies, research and work. Hence the usual STEM -> patent law thing isn't really there.
3. In quantitative circles, research positions at hedge funds (e.g. DE Shaw, Citadel, Two Sigma) are more coveted than top tier software engineer and banking analyst roles in that they pretty much require a PhD in maths/stats/CS/physics/EE (unless you're some kind of genius). But most people haven't heard of these funds. I'm worried that adcoms won't respect my company (its one of the above) as they would respect someone from e.g. bulge brackets, Google/Facebook/Amazon, McKinsey/Bain/BCG, which is rather annoying as I had job offers from Google and Goldman among others before starting my PhD. At that time I had been rejected without interview for every single role comparable to my current job!
4. Main one: my softs are boring/unoriginal.