Why does everyone assume that all the Pokémon are fully trained, level 100, with perfect movesets tailored for mass murder? Trainers are not in the consideration. It's "billion lions vs all pokémon", not "billion lions vs well-prepared Trainers and their Pokémon". No Leppa Berry shenanigans.
1, I was assuming level 50. And given real-world abilities, I was being GENEROUS modeling lions as level 40 Pyroar without special attacks
2, quite a few Pokemon require trainers to evolve. They are included. Therefore, they have some training. I didn't assume trainers would intervene; Psychic types are fine for the strategy components
"PP" means "you can't spam Flamethrower for hours and hours endlessly, the mon will get exhausted and overwhelmed". The battle will last for a very long time and will be way more exhausting for the Pokémon who will have to defend all the time, while most of the lions will just wait and do nothing.
Due to size constraints, lions only field a part of their power at a time. Pokemon retreat and recover while another pokemon covers the front line.
Using most moves (like Flamethrower for example), the Pokémon makes itself vulnerable to the hundreds of lions attacking it. Lion is not a weak, vulnerable animal either, defensively they should be easily compared to most Pokémon. They can withstand a few attacks and they absolutely can hurt the Pokémon with their own attacks. A Loudred won't just easily kill a thousand of lions just because it learns Hyper Voice, it will hurt - not kill - like.. thirty.. fifty.. lions that are standing in front of it before the others get behind it and tear it apart. Regular multi-target attacks won't hit every lion. They will hit a ton of them sure, but again, "hit" does not mean "kill", they can backfire and hit other Pokémon, you have to hear the Perish Song for it to kill you, etc.
I checked the damage calculator. Being GENEROUS, they still can't survive. I plugged in a generous level 40 lion (pyroar) against a few (STINGY to pokemon) seasoned-but-not-even-close-to-max-level Pokemon. It falls in ONE HIT.
Standard biting attacks are Dark-type, thus lions can hit Ghost-types. Pokémon also are not unable to damage the Steel-types with claws and teeth.
double typeless - both lion type and attack. I'd argue they should be normal types, but with Bite/Crunch, that doesn't really matter, especially since all they do is scratch damage (a few % of hp to the best, who can fall back and recover)
No Flying-type Pokémon can fly endlessly and i doubt your usual Pidgey or Staravia are powerful enough to eliminate *that* many lions before they get tired, land and get torn apart by the lions.
Rayquaza says hi. Also Drifblim and Drifloon. Bats like Woobat/Swoobat can also sleep on ceilings that the lions can't get to. And Sigilyph is an automaton that goes for thousands of years.
The question eventually becomes - "who would win in a fight - 990 millions of lions or the few legendaries with actual powers?" Ignoring the Pokédex fairy-tales which clash with everything shown in the anime, games, everything - lions will win.
There's more than enough in OU who're powerful enough to put a dent in them. A very large dent, with others supporting them. And besides, the demonstrated powers of some of the legendaries is more than enough. Ho-oh says hi with resurrection... multiple legendaries don't even need to be on the field to fight... many more legendaries are, when not restrained by a pokeball, OUTRIGHT APOCALYPTIC.
Speaking of Pokédex, just please try to imagine either Pokémon in real world or lions in the Pokémon world, with mechanics of the respective world applied to everyone. An even field, you know.
Yes. It's called Pyroar. But scaled to humans. Frankly, the level 40 I gave them was probably TOO MUCH compared to most pokemon. And they still lose to the pokemon with that.
Pokemon in the real world:
Basically, the legendaries make it not even a question. Multiple simultaneous apocalypses, some not even triggered from on earth, lions dead, pokemon win. Humans are also probably dead in this scenario. And the lions can't concentrate their power properly to stop them all; only a few will fit in the space to attack in the first place.
Lions in the Pokemon world:
Then Focus Sash and the like are not things that are common enough for them to get, and even so, that'd just drag it out. It'd be a HUGE slugfest, but eventually, the lions finally fall, ground down to nothing.