I agree with the last post. Assuming the flights were all on one TATL itinerary , the OP should have been allowed to check 2 bags without an additional charge.
The rule as published states: "For international customers making a connection/stopover to or from the U.S., or at a later date during their travel within the U.S. on the same itinerary, international fees and rules apply."
DL Medallion members may check 2 bags without charge but the same rule does not apply to the SkyTeam partner flyers. That rule states: Will SkyTeam Elite/ElitePlus customers be given an exception to the charge?
No. SkyTeam Elite/ElitePlus customers traveling in the economy cabin will be subject to the appropriate checked baggage fees.
The OP has a valid point. If the Skyteam travel is supposed to be a seamless hassle free experience, there should be consistent rules on luggage and lounge access.