The Brotherhood of Steel is not the power that we once were. We believed ourselves to be the sole source of technology left to mankind. Secure in this belief we have let our order decline over the years.
He gives the Chosen One the quest about Vertibird plans when they first meet.
If the Chosen One completes the task, Matt ends up being killed by the Enclave's own Frank Horrigan 2 days later (It has not been confirmed what the Chosen One's decision was, or if Matt survives, in Fallout canon).
The Chosen One also gets to keep the original plans for use with the other quests from the Hubologists and the Shi. Two days after this Matthew will disappear, and accessing the ACE computer in the bunker will reveal what happened to him by playing back internal sensor recordings.