You've hit on yet another area where I think the NEC needs a bit of editing.
I don't think the committee ever intended to say 'you can only use a box that is specifically made and marketed for fans.' After all, we have decades of safe fan installations that predate the code requirement.
Indeed, it seemed almost ridiculously obvious: an inadequately supported fan would promptly fall, and the guy who hung it would get to fix the mess and try over. As you learned the trade, you learned how to make sure you didn't get THAT call-back.
Then a few firms came out with clever boxes that let you retrofit a fan-strong box in places where you didn't have access to the back side. Someone (naturally) asked for listing, or some assurance that the gadgets would perform as advertised.
Another eagle-eye noticed that the traditional 'pancake' box was a bit shy on wire space; so fancy boxes were invented.
The next thing you know, we have code language that requires you to use one of these purpose-made boxes.
I'm sorry, but a nice 4-square box, screwed to a solid piece of lumber that you have attached to the ceiling framing, is plenty strong for supporting a fan. I suspect the box / bracket you describe is also plenty strong. You just lack the testing and listing that the current code requires.
As much as I like complying with codes, I'm not sure if I have ever used a 'listed fan box.' I doubt I ever will. Except for a few special circumstances, I just don't see the need. Then again, I never imagined that someone would try to hang a fan without using any box - let alone a flimsy plastic 'old work' type.
It seems that as soon as things are made 'idiot proof,' they come out with a better idiot.