I'm fairly certain that Delta's main issue with the Ex-Im is that they provide subsidies to state owned/run airlines which tend to already have large advantages. Providing them a way to obtain cheap aircraft and then jam a bunch of capacity into a O+D can do some serious harm to US carriers.

Take a look at Air India - they have had all kinds of financial problems but got some money from the Ex-Im and killed the markets for NYC-BOM and NYC-DEL by introducing more capacity and putting out some super cheap fares. There is a reason only UA is the only US carrier with a non-stop to Inida from NYC.