Hitmonlee DOES have a niche, you're just trying to use it as something it isn't.
It is one of the best anti-offensive weather pokemon in the entire game.
Calling Hitmonlee "the best" anti-weather Pokemon is
a lot of wishful thinking.
If you find yourself against Sandstorm, then Gliscor will wall Hitmonlee for days, render the effects of his ability wasted, and set up. Landorus also laughs at Hitmonlee and Reuniclus gladly sets up on it. Skarmory doesn't care about Blaze Kick seeing as how Hitmonlee can't carry Life Orb with Unburden.
If you're in the rain, Rotom-W can take on Hitmonlee, seeing as how his attack is not threatning because his ability requires him to forgo Life Orb. Jellicent will wall any attack from Hitmonlee. Dragonite can set up while utilizing Multiscale. Jirachi in the rain won't care about Blaze Kick and will set up, and the same goes for Celebi. Starmie will run rampant. Toxicroak will shrug off its attacks and set up.
Pokemon that are found on both Sandstorm and Rain:
A Choice Band Scizor will 0HKO Hitmonlee with a single Bullet Punch 110.7% - 130.6% of the time. Lati@s aren't threatened and could set up CM/Sub if they wish. Gastrodon and Vaporeon will status Hitmonlee and absorb its blows.
Hitmonlee doesn't belong in OU. It's much easier to list out the OU Pokemon that aren't threatened by Unburden Hitmonlee than to actually list out what Pokemon you're beating that warrants Hitmonlee. When you consider a Pokemon for your OU team, you can't just say "so-and-so is fast and powerful so I'll put it in". You have to think about synergy and whether the Pokemon can even compete in that environemnt. Using Hitmonlee in OU hurts you because you're losing out on using other Pokemon that can do much more beneficial things for your team.
Has anyone considered using a Cell Battery with Unburden, or is it too unreliable for competitive play?
So you'd rely on predicting an Electric attack and hope that Hitmonlee survives it just to get a speed boost on a now damaged Hitmonlee without any item to boost its attacks? Does that sound like a good plan? And if you switch Hitmonlee in on a Volt Switch, your opponent will switch out to something that walls Hitmonlee, thus rendering your ability and item worthless.
Forming a competitive team is about synergy, not just throwing together Pokes in hopes that one of them will sweep (which Hitmonlee can't do much of seeing as how it can't even hold Life Orb).
I don't mean to have an animosity against Hitmonlee today, I just don't have the same optimism people seem to have about Pokemon single-handedly sweeping through teams. Today's metagame doesn't make that very easy nowadays.