it’s simply common derech eretz. Rav Freifeld was very makpid regarding this, see his biography (a true mussar sefer).
“shul is not mine or yours, but Hashem’s” we have an obligation to emulate the Ribbono shel Olam, v’halachta bidrachav. He gives sholom to us-sholom sholom l’rachok ulkarov! in fact the rachok, maybe a guest or stranger? (as well as baal teshuva) gets sholom first!