The street segment shows as private access in two layers (Standard & CycleOSM) (which is correct; there is a gate restricting access on either end). However, it does not show as private access in the Cycle Map or Transport Map layers. Should it? What’s the way to make that consistent across all layers?
The info (data) is basically all consistent, except for delays from different update frequencies, or how it excerpts the database. The visuals of different renders is expected to be unique. That’s their raison d’etre.
The ones now are all raster, not easily directly customizable vector maps. If you don’t want to render a copy of the entire style, you could still eg overlay access= lines on existing tile services, using it as a basemap, which is what most of them are intended as.
Why should a transit map render access=private ? There’s only a need if you give equal priority to walking and bike for last-mile and transfer connections, but then you can overlay a transit layer over other basemaps instead.
Thunderforest Cycling is more focused on official routes and dedicated infrastructure. CyclOSM has more provision on sharing other roads. CyclOSM - OpenStreetMap Wiki
What’s the way to make that consistent across all layers?
It is impossible in general as different layers may do rendering using own rules that may change at any time.
In general it is recommended to add accurate data, in formats usable by data consumers.
To get advise specific to this case: please share location or at least exact current tagging.