From this list you can tell that no rain check covers everything on a rain team, and the only way you can consistently beat rain with one rain check is to either prevent rain from going up at all with some hax shenanigans or running multiple fast taunters, or hope you can make exactly the right switches for 8 turns.
And you can't run every Swift Swimmer on one team.
Toxicroak: Ohko'd by +2 Stone Edge from Kabutops. Kabutops survives LO Vacuum Wave and obviously survives Sucker Punch. Ohko'd by Zen Headbutt from Physical ludicolo, Psychic from Gorebyss, and Earth Power from Omastar. Qwilfish can just blow up on it.
Jolly Kabutops: Ohko'd by +2 Aqua Jet. Doesn't ohko Ludicolo, Omastar, or Gorebyss while obviously being ohko'd back. Qwilfish is faster and kills it before it can do anything.
Who's going to let you get to +2?? People arguing for Rain are all like "+2 in Rain means you lose". Setting up Rain, switching to sweeper, Swords Dancing
takes 3 turns. Even switching Venusaur into U-Turn lets you deal with at least 1 Rain sweeper, since none can OHKO unless you're running Specs Gorebyss or something (Which I'm going to switch out, since you're not setting up). Basically, consider, Rain goes up, Uxie uses U-Turn, in comes Venusaur. Kabutops comes in. Do you Swords Dance up or try to kill Venusaur?? Either way, Kabutops is losing. All the while, 3/8 rain turns is already up. If you're playing with your old style, you go back to Uxie and back to Kabu, that'll be like 6-7/8 teams gone. The person defending against Rain will already be preparing to curbstomp you when Rain stops.
Ultimately extremely good players can beat other extremely good players using rain only by significantly tailoring their team to beat it; however, it's often the case that adding multiple rain checks is not a bad thing because they're good for other things besides rain (as evidenced by Toxicroak usage skyrocketing and staying high).
This is very true. There are also Pokemon who aren't "specific Rain counters" that can check Rain should the time arise. Using my RMT as an example, none of the Pokemon on there are specifically used to beat Rain, but it defended against it very well.
Kabutops lead: Can kill slower, weakened Rain sweepers
Drapion: Toxic Spikes, also bulky enough to take physical hits and phaze them out
Tangrowth: Walls physical Rain sweepers
Rotom: Can revenge slower Rain sweepers like Gorebyss
Hitmontop: Can Mach Punch Kabutops, Sucker Punch (You can't stall out Sucker Punch either, since Rain is running out beforehand) others and stuff like that
Moltres: Comes in on Rain Dancers and Subs up, leaving you helpless
So even with just a team that isn't specifically tailored to beat Rain, it can still easily beat. Specifically tailored teams would fail, since they would lose to everything else.