Storm door panel replacement

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".

The closer has a few purposes:

• it closes the door so that you don't have to remember to every time. It will also pull the door closed if it opens accidentally (for example, if the wind catches the door).
• it prevents the door from moving too fast, and usually prevents it from opening past about 90 degrees. (Anecdote: I know someone who didn't have a closer on his storm door. The door opened so that it would catch the prevailing winds -- which are often strong -- in his area, but he never had problems until one time he forgot to close it and it swung open, hit the wall next to the door and knocked out a pane of glass. With a closer, it might not have been able to move fast enough or far enough to do any damage.)
• it provides a second point of attachment to the door frame for the side of the door that opens, which helps to hold the door if the wind catches it.

Does the door close securely, and will you always remember to close it securely? If not, I'd say you need the closer.