I can add one more example from geometry, topology and physics, where simple modular Lie algebras arise. In John Milnor's article On Fundamental Groups of Complete Affinely Flat Manifolds there was an open question for Lie algebras $L$, namely which Lie algebras admit a bijective $1$-cocycle in $Z^1(L,M)$ with $\dim(L)=\dim (M)$. For the adjoint module, this asks for a non-singular derivation. Jacobson proved that in characteristic zero such a Lie algebra must be nilpotent.
The question was formulated there also for Lie groups, namely which connected Lie groups admit a left-invariant affine structure. It can also be formulated in terms of Yang-Baxter groups, or left braces, see the article Counterexample to a conjecture about braces by D. Bachiller.
From the first Whitehead Lemma it follows that a simple Lie algebra of characteristic zero does not admit such a $1$-cocycle. In prime characteristic, this is no longer true. For modular simple Lie algebras there are such examples, see First Cohomology Groups for Classical Lie Algebras by Jens-Carsten Jantzen.
This has then interesting applications to $p$-groups, Yang-Baxter groups, left braces, pre-Lie algebras, post-Lie algebras, and other things.
Benkart, Kostrikin and Kuznezov classified Finite-Dimensional Modular Simple Lie algebras with a Nonsingular Derivation.