He didn't even demonstrate his ability. How are they supposed to believe him...
They probably skipped parts of the discussion, but assuming they didn't.
The entire discussion 100% hinges on a dark elf "slave" telling the truth to them while not under direct supervision.
That said, the question about her being happy indirectly tells them quite alot.
There really isn't anything to strengthen their case other than their word and the fact that they made it this far.
MC is being EXTREMELY forward with his request, but he was honestly right about telling them about her slave status.
Considering the story beats so far....
I'm betting that if they do agree to the idea because their current situation is shitty and its worth the risk.
They are effectively dying out anyway if you look at the living situation and from what was said...
But if not, he could take somebody there and back to prove his ability and the fact that he is telling the truth.