Doors – Stain Grade Door – piece of veneer missing – how best to repair

I agree with Jeff Widmer -- you don't need a bigger/stronger bracket, as it's not the bracket that failed, but longer screws to properly fasten the bracket.

Given that, your jamb repair doesn't need to be structural so don't sweat how strong it is and just make it look nice with whatever product you've used before and liked. After you drill your pilot holes for the new bracket screws, drill out the repaired jamb with a bit that's larger than the screw threads so the screws don't stress your repair. Of course you want this larger hole to only be the depth of the jamb, maybe 3/4".

Unfortunately, I'll be surprised if you can effect a very successful repair. Restoring the outer corners like those is tricky at best. And having it stained only makes it worse, since the wood grain will be easily visible.

Most likely any attempt to repair will make it worse looking not better. I'd opt for something low-impact, such as a scratch-cover product like Olde English, which essentially re-stains the now-exposed wood fibers. It at least conceals the bright colors of the underlying wood and makes the damage a little less noticeable.

But ultimately I'd confess to your landlord and let her decide how best to proceed. It's her door, and her call.