This self-answer shows the solution I came up with:

This is sort of a simpler version of some ideas suggested by isherwood and jay613 -- thanks! This didn't even require a trip to the hardware store. I just had some scrap lumber, nails, and screws around. The only tools I used were an electric drill, a hand saw, a screwdriver, and a hammer.

I'm still not sure if it's strong enough to stand up to a once-per-year wind, but the white bracket is now mounted onto flat wood, which is held in place by a bunch of screws and nails. I think if I were to grab the flag and pull down hard, the longer lever arm would certainly win, and the whole thing would come apart. To make it stronger, I think the thing to do would be to add some kind of shim in the air gap below, and/or a steel strap wrapping around on top...?

This self-answer shows the solution I came up with:

This is sort of a simpler version of some ideas suggested by isherwood and jay613 -- thanks! This didn't even require a trip to the hardware store. I just had some scrap lumber, nails, and screws around. The only tools I used were an electric drill, a hand saw, a screwdriver, and a hammer.

I'm still not sure if it's strong enough to stand up to a once-per-year wind, but the white bracket is now mounted onto flat wood, which is held in place by a bunch of screws and nails. I think if I were to grab the flag and pull down hard, the longer lever arm would certainly win, and the whole thing would come apart. To make it stronger, I think the thing to do would be to add some kind of shim in the air gap below, and/or a steel strap wrapping around on top...?

But anyway, at least for now, with the help of stackexchange, we are once again flying the flag that to me and my wife symbolizes the constitution, the Bill of Rights, and the war that our grandparents fought against fascism.