This is a great question, especially the part about the OSCE's role.
We have that organization involved and also the OSCE Minsk Group. The co-chairs are France, Russia and the U.S. As you probably know, the Minsk Group has been facing enormous problems in resuming its functions after the 2020 war due to Azerbaijani resistance to engaging with the group. Baku believes the group failed with the negotiations regarding a peace accord. Baku still tried to engage with the group, but then they saw that it wasn't really going the way they wanted and the process got into a deadlock.
The other reason we're not seeing the OSCE Minsk Group come together is the Russian invasion of Ukraine. When the west got much more involved in mediating between the Armenian and Azerbaijani leaders, we saw Russia basically withdraw from the group. It's still there on paper, so there is potential for it to come back, but at this moment, it's difficult to see any kind of co-operation taking place between the U.S. and Russia, including on Nagorno-Karabakh, unfortunately.
Having said that—