Originally Posted by MSPeconomist

This seems wrong. As a DL 3MM, you're automatically GM which is SkyTeam ElitePlus. As such, you have access to all SkyTeam operated business class departures lounges when traveling the same day on a SkyTeam international itinerary.

The Plat AmEx card doesn't get you into AF lounges (unless they're PP and you show your PP card), but your status and ticket should. Your elite status should be printed on the boarding pass, so if the agent had looked, it should have been obvious that you were entitled to enter. Similarly, a DL or AF business class boarding pass should suffice for AF lounges except the special IFC AF La Premiere lounge at CDG.

The AF arrivals lounge landside at CDG (which AFAIK is still closed) has somewhat different rules.