Note that LA is not a Skyteam member, so I don't believe DL STE+ had access to DL Sky Clubs prior to changes when flying LA (only access to the 5 LA operated lounges due to more limited relationship). While a Sky Club membership allows for access when flying any partner (except HA which must be DL marketed), DL STE+ status only notes access previously to DL Sky Clubs when flying actual Skyteam partners -- https://www.delta.com/us/en/delta-sky-club/access