I've been doing SPA's in react for a while and have had 0 problems whatsoever with styling. As long as you stick to good standards and naming conventions and use SASS or Stylus or something to keep things organized.
What tends to give people so much trouble with styling?
Isn't that part of the React mantra? Everything is completely unopinionated except for rendering HTML based on data? Like it makes no opinion on how you should do state or HTTP requests as well as no opinion for styling, which is something I love since it avoids duplicating code or incompatibilities with standalone libraries.
What tends to give people so much trouble with styling?