I can't speak for "the left", whatever that is...
...but I think:
- The USA (paradoxically, considering all the bad things you hear about us) enjoys a reputation as the only intermediary that most parties in the world will listen too. We have diplomatic "star power".
- When communications break down between two parties, intermediaries are vital.
- We have a huge vested interest in there not being an India/Pakastan war.
- Serious diplomatic efforts should not be confused with sabre-rattling public statements. It's ok and consistant to favor the one and criticize the other.