I realized I confused the highway=busway tag with the deprecated service=busway tag and concluded that being the reason for the lack of rendering/routing in OSMCarto, and also further connected the dots with some other busways I've seen being tagged with the highway=service + bus=designated scheme, so I switched them to the one I'm more used to seeing and under the belief that it was the replacement for the deprecated tag. Turns out the other tagging scheme for busways is plain incorrect too, so I've restored the traditional busway tags here.