We explain how to interpret the complexes arising in the "classical" homology stability argument (e.g. in the framework of Randal-Williams--Wahl) in terms of higher algebra, which leads to a new proof of homological stability in this setting. The key ingredient is a theorem of Damiolini on the contractibility of certain arc complexes. We also explain how to directly compare the connectivities of these complexes with that of the "splitting complexes" of Galatius--Kupers--Randal-Williams.