Thx for clarifications,
From what you are saying, it seems that the DM does not actually have a graphical use? - That it only starts the graphical session but for the most part has no significant effect on the graphical environment (except the well known SDDM lack of X server support I describe again below. X server is required for remote connections, but not for locally running graphical environments today)?
The “IceWM” login page may not exist, and likely displays something that OpenBox provides. It only has login fields User and Password and unlike “proper” Desktop login pages does not offer other functions like Desktop choice.
What you wrote still does not change the fact that no matter what WM or DM is chosen, the login page’s appearance will not change, so according to your description is probably a bug… yet, as many can verify <functionality> choosing for example SDDM vs LightDM has a big effect on whether X server for remote connections is supported.
TSU