Fixing brick cracks above garage door

The idea of pushing a brick veneer wall back onto its bearing surface is really not a good idea. Besides water will not push it off, something else did in my opinion, like wood that swelled up from the leak, but that will be minimal. I would still be more concerned about the water around the framing behind the brick. If the brick ain't fallen yet, it may not. But it may need to be taken down to repair the band joist, plate, perhaps. Then set the brick back with proper waterproofing in place so when water gets behind the facade it won't matter. FWIW, in a building science seminar I attended, the speaker Joe Lstibureck, discovered that brick allows 10 percent of the water that hits/saturates the surface to pass through its thickness to the what is a hopeful moisture barrier behind the brick, in front of the framing...

From studying your picture it appears to me that the timber is intended to tie one wall to the other. If so, I would be asking why. A couple of clues I would look for are:

1. Is the height of the placement of the timber, on each end, possibly where the top plates of the framed wall are, behind the drywall? ( As opposed to the studs inside the wall running from the floor plate all the way up to the roof rafters.) If so, this connection between lower studs and upper studs could be a weak area for lateral wind loads to cause some unwanted flexing of one of the walls.

2. You mentioned your front door was on the other side. Is it possible that the front door wall, beyond, is located in a sturdier wall that has sufficient bracing and the timber in your garage is using the strength of that wall to tie the opposite garage wall into something more stable. If that opposite garage wall is an exterior wall subject to wind I would think that might be the case.

There are some DIY things you can do to experiment with it a little bit. Try flexing the timber back and forth to see if one wall stays rigid but the other moves just a tiny bit. If so, that would indicate that the wall that moved needs that timber.

A good framing carpenter ( think This Old House) could probably access your situation adequately. Maybe for free if it is easily apparent to them what is going on and they can give you an estimate for any modifications that would enable you to remove the timber.

Hope this helps. Good luck!