No. It's not my problem. It's not our problem. The endless guilt trip ends here. We are not obliged to take a single soul from these countries. Our only obligation is to leave those areas the hell alone and look after our own civilization.
They have radically different ideas than us on how society should be run. That means both mass immigration from those countries and our own attempts at spreading liberal democracy to them are both equally foolish.
And what will the UN do if we chose not to comply because the original intent of said convention is now being abused as an excuse for mass third-world immigration?
No, I'm serious, what will the UN, a cabal of unelected bureaucrats who I have never voted for, do to my country if we don't comply with said convention?
Treaties like are predicated on the will of the signatories to sanction each other for non-compliance, which can take subtle or overt forms. While that by itself is no guarantee of change, it may be that you have underestimated the depth of others' political feelings on the topic.
They have radically different ideas than us on how society should be run. That means both mass immigration from those countries and our own attempts at spreading liberal democracy to them are both equally foolish.