For what you're describing, I think differential geometry fits the bill very well. Can be pretty abstract and hard to get into, may be very technical, but I doubt that most people would think of it as the height of abstraction in maths.

Vast applicability in physics - general relativity, gauge theory, classical mechanics and continuum mechanics usually forwarded as the crown jewels. Lie theory and representation theory. Geometric analysis, in which we've seen some of the most impressive PDE methods with stunning results in the past few decades, especially relating to topology. Complex geometry, where there is particularly rich interaction with algebraic geometry.

Dynamical systems. Optimization on manifolds. Discrete differential geometry, which is of interest in computer graphics and design, as well as in combinatorial surfaces I suppose.

I've seen differential geometry getting talked about in the context of machine learning (manifold learning?). There's apparently stochastic differential geometry as well (think Brownian motion on Riemannian manifolds).