Do you mean that https://safecyclingmap.com/ isn't using Mapbox? Because I see the logo at the bottom right of the website.
it uses mapbox'es renderer (mapbox-gl) and seems to pull in the CSS from mapbox directly. Might be worth changing to maplibre since it's not using mapbox'es tiles.
It uses Mapbox GL JS v2.x, which is no longer open-source and is tied to the TOS, so the distinction of using Mapbox tiles or not doesn't seem to matter. Just loading the library requires a token from Mapbox.