Off the top of my head, they could have had lucio ball or other summer/olympic themed modes cycle for the 3 weeks and create some kind of competitive ladder similar to the one thats incorporated in ranked and depending on what tier you break into (top 500, top 1000, top 5000, etc) you get some kind of designated reward package that you'd otherwise would have earned. The only problem here is that it would have created a completely different player pool/queue which i know they don't want to go overboard with but being it's a timed promotional event i'm sure they could deal with it. This is why no one complains about the unlocks/rewards tied to competitive play; it's clear what needs to be done to get what you want and everyone has a fair shot.
That's what makes the system all the more baffling and insulting, they have a pretty good set up with the way rewards work in ranked. So clearly they could have created a better system but chose not to.
The content is rare because the system it works within is unfair and is based on nothing but luck and RNG
I'm suggesting rare content and a fair system to get said content needn't be mutually exclusive.
I think it's weird that you think RNG is "unfair" and want to replace it with a system where some people will have an inherent advantage. In both cases, people with more free time are at an advantage, but you want to layer on another advantage to certain people with skill. Your solution makes the items even less obtainable for certain people, so in a way it's less fair.
I guess you could argue the current system also gives an advantage to people with money to blow, but that wasn't a reason you cited for why it was unfair -- and unless your solution also allows for people to spend money to at least increase their chances (so it'd have the same unfairness), it doesn't address Blizzard's goal of generating revenue to support continued development.
It's worth thinking about the place these items fill. You'll notice that the rewards for competitive play are designed as prestige items. It's less about personal expression and more about displaying accomplishments. It's just a gold weapon -- it's little more than a trophy. That's in contrast to skins and sprays etc, which are more a form of expression. It doesn't seem like Blizzard wants to limit those kind of expressive skins to certain skill ranges -- maybe they think it would demotivate less skilled players and make them more likely to quit because they assume they'll never get the item they want, whereas the current system gives them some hope because it's random and not based on skill -- they have the same chance as anybody else.
And yeah, I think a solution that draws people away from playing the actual game is a pretty poor design. Brawls are intended to be side attractions -- incentivizing playing them over the core game goes against that ethos.
edit: this is why I like valve's approach where there's a marketplace for items and players can sell/trade. Allows items to be rare, but as a player you can go after specific items more efficiently.