Lin's theorem shows the existence of a localized basis for the low-energy space in models of non-interacting fermions on a finite lattice on a disk. This was observed by Matt Hastings.
See "Topology and phases in fermionic systems" in the Journal of Statistical Mechanics: Theory and Experiment, 2008, L01001, especially the next to last paragraph.
Other norms come up in applications. It can be hard to figure what norm is relevant in engineering and science papers, but it is often clearly not the operator norm. Generically speaking, approximating two commuting hermitian matrices by commuting hermitian matrices is equivalent to joint approximate diagonalization of those matrices.
A famous algorithm in blind source separation if by Cardoso and Souloumiac, "Jacobi angles for simultaneous diagonalization" in SIAM Journal on Matrix Analysis and Applications, vol 17, no. 1, 161--164, 1996.
The Cardoso and Souloumiac algorithm was used later in computational quantum chemistry, as in Francois Gygi, Jean-Luc Fattebert and Eric Schwegler, "Computation of Maximally Localized Wannier Functions using a simultaneous diagonalization algorithm,"
Computer Physics Communications, Volume 155, Issue 1, 1 September-15 September 2003, 1--6.
Jon von Neumann considered almost commuting operators. See "Proof of the ergodic theorem and the H-theorem in quantum mechanics," The European Physical Journal, 1--37, 2010 and the commentary that accompanied the translation, "Long-time behavior of macroscopic quantum systems, by Goldstein, S. and Lebowitz, J.L. and Tumulka, R. and Zanghi, N., The European Physical Journal H, 1--28, 2010.