I don't know about Award Nexus, but when EF shows availability in the right code, it's supposed to be inventory that is available to SkyTeam partners, not just for FB miles.

OTOH, the same does not seem to be true for KLM: EF and even delta.dumb can show TATL availability on KLM operated flights, such as AMS-ORD, but DL agents either cannot see it or cannot book it, even when they claim to try doing a long sell. Trying to book these award tickets using a KLM segment directly on delta.dumb results in repeated error messages.

If Award Nexus, KVS, etc. are getting their award availability data directly from the AF website, that presumably includes all seats available for FB miles, some or all of which might not be available using DL or other partner miles. (There might be additional AF award inventory available only to FB elites when they are logged into their FF accounts, for example IFC award seats.) My understanding is that when you see 9 (low) award seats in the correct inventory class for AF flights on EF, they should be seats that AF is willing to make available for partners. Of course, there could still be a question whether DL is willing to let you use (low necessarily) miles to get those award tickets.

This phenomena became known as *Net blocking when UA was doing this: not allowing UA customers to use UA miles for partner flights that their partners were willing to make available to UA and other *A FF programs. I'm not aware that DL has ever been credibly accused of doing this. IMO the evidence is that they did not do this for SQ award tickets, even at the very end of that bilateral partnership.