Beartic is amazing in rain.
About Fighting-types, you are exactly right - which is why I stated right away Leavanny should never be your only Fighting-check. Ever. Leavanny just deters their STAB as well as Earthquake coverage, which can allow you to do things otherwise impossible, like staying in with a Steel-type (if you know that Fighting-type is choiced, for example), or give an already solid check an easier time checking (Misdreavus for example, does not like taking Banded Mold Breaker Earthquake from Sawk, but Zen Headbutt or Ice Punch are way less of a problem.
About Psychics, Leavanny can actually switch into most if it really needs to, but indeed it is advised you don't. But neither can Pinsir, for that matter, or even Scyther because Thunderbolt and Thunder Wave are a thing, and Jynx has STAB Ice Beam.
Also, there is this... If Leafeon is your only Water-check, you won't do well (not that Leavanny should be your only water-check!).
216 SpA Lanturn Ice Beam vs. 0 HP / 0 SpD Leavanny: 160-190 (54.9 - 65.2%) -- guaranteed 2HKO
216 SpA Lanturn Ice Beam vs. 0 HP / 0 SpD Leafeon: 190-224 (70.1 - 82.6%) -- guaranteed 2HKO
Boltsandbombers actually gave much better reasoning; if you can't switch in, you won't be listed. But if we go with that reasoning, remove Leafeon from Water-type checks as well because it dies to every remotely powerful Ice Beam.
I wouldn't be so quick to jump the gun on pointing out others being wrong if I were you.
As far as I am concerned there is literally no reason to use Leavanny as a physical sweeper over the likes of Sawsbuck or Leafeon, which are faster and more efficient attackers. If you want Bug moves so badly you could always try Megahorn Sawsbuck, and Sawsbuck has the benefit of powerful Normal STAB. There's also the fact that both Sawsbuck and Leafeon have access to Natural Gift, which is a very important advantage as those two basically have an additional coverage move that Leavanny does not. Scyther and Pinsir are also superior offensive Bug-types due to their stronger neutral coverage, while Scyther has bulk, Speed, and Roost on its side while Pinsir has great neutral coverage and Mold Breaker to bypass Quagsire without the need for Grass STAB. The team in that replay you've shown succumbs to Sawsbuck and Natural Gift Leafeon all the same; more importantly they'd do it
better since Sawsbuck's healing and Leafeon's physical bulk would allow them to shrug off Kangaskhan's priority much better than Leavanny.
Funnily enough, I never disagreed with others being wrong - they are not. You call Scyther and Pinsir better Bug-types, which is the truth. But they aren't Grass-types as well. Similar for calling Sawsbuck a better Grass-type than Leavanny, which is also true - but it isn't also a Bug-type. And Natural Gift is cool but Leavanny has 4MSS as it is; it already carries 3 attack moves, and it still wants Webs, Heal Bell and Synthesis (and possibly Magic Coat) as well.
There is also this, an example of where that Bug-type comes into play:
4 Atk Malamar Superpower vs. 0 HP / 0 Def Sawsbuck: 216-256 (71.7 - 85%) -- guaranteed 2HKO
4 Atk Malamar Superpower vs. 0 HP / 0 Def Leavanny: 48-57 (16.4 - 19.5%) -- possible 6HKO
252 Atk Leavanny X-Scissor vs. +1 252 HP / 0 Def Malamar: 336-396 (89.3 - 105.3%) -- 31.3% chance to OHKO (Guaranteed OHKO with Life Orb or Insect Plate, or SR on the field, but you often enough lose your item because you switched into Knock Off instead).
Non-X-Scissor Leafeon is also set-up bait for Malamar.
It means indeed nothing, but you missed my point; every mon with an unique typing (as as you said, the other two are irrelevant) has a niche. Always, through virtue of said rare or unique typing - as long as it has the tools to use their typing, that is. There will be mons no one has discovered the niche of as of yet, or sets we don't know about but mon X can use it solely because it has that beautiful, gorgeous typing. A typing that might suck for all other purposes, but yet still grants it that tiny niche, or that ease in teambuilding in covering things up for certain players.
About Leavanny and Gourgeist; They check them in different ways. Both are Grass-types that resist Fighting and share a bunch of weaknesses. Gourgeist will stall them out with Seed and WoW, whereas Leavanny will kill them. Gourgeist has far more staying power, and will stay on the field much longer. But both are used to check and/or counter Malamar, Water-types, and Ground-types. You exchange a better match-up versus Fighting-types (Gourgeist) for a better match-up versus Water-types (Leavanny) but they use their typing to check things. It seems odd at first, but due to their entirely different functioning, their partially shared typing and the things their secondary typing hits for SE damage, and the fact they share many weaknesses as well, you really will never use both on the same team unless you go Kamikaze with Leavanny.
ou use them entirely differently, sure. But you'd be surprised at how similar the things are they check. Obviously there are differences, for example, Gourgeist checks Primeape whereas Leavanny checks Samurott, but there are many things they both check.
A long while ago, I had a team that was weak to Heatmor but not to Magmortar because it has Sucker Punch and Giga Drain - when Seismitoad was my Fire-check. Oops.
Also, why mentioning Shiftry as a reason for Leavanny's downfall? Shiftry's rise is only a good thing for Leavanny as Leavanny is a solid check to it.