Allow me to be gentle.
Reading documents from Rome is not quite the same as reading speeches by Senator Blowhardfoghorn.
Rome is usually VERY precise in its use of the language. That means that what they said is what they meant, no more, no less.
When JPII says "...based on [the experience of] the UN.." it is NOT a compliment to the UN--it is a slap.
What that really means is that 'the UN is a good idea, sometimes actually efficacious, but not nearly often enough...'
Even if you read his speech as an endorsement of the UN, please recall that the UN's Sec.Council has the USA as a Permanent Member with veto power.
Maybe that's what JPII wants to preserve...?