Question

What are the assumptions behind “term symbols”?

Answer 1

In multi-electron atoms, the electronic state of the optically active " subshell" is often expressed in "term symbols"  ...

Answer 2

Term symbol $^{2S+1}L_{J}$ is a collection of approximated good quantum numbers. The assumptions may be rephrased as approximations. As you mentinoed, for Dirac equation, $L^2$ nor $S^2$ does not commute with Hamiltonian even in hydrogen atom (ref. Advanced Quantum Mechanics by Sakurai). Other approximation like single-electron picture does not appear, since $[H,L^2]=0$ anyway. You may image the coupled angular momentum is a generator for rotating all electrons together, which commutes with the Hamiltonian. Cohen-Tannoudji's QM vol II, X.2 has a excellent discussion for this point.