I don't know the full details when it comes to doubles but in singles, the reason you don't run multi weather teams is because weather teams themselves have to follow a certain team structure. If you don't follow that structure, you will be destroyed by a Pokemon who can threaten your weather team
For example, let's take a singles rain team. Those teams would usually consist of Pelipper, Barraskweda, Ferrothorn, Seismitoad and Tornadus. The last slot can either be a thunder user like Tapu Koko or another water type attacker like Volcanion. It is very rare to see a rain team deviate from that structure because each of those mons contribute to the sturdiness of rain. Pelipper is obviously needed for the rain and Barraskweda is your speed control. Ferrothorn provides the mandatory steel type while Seismitoad is to protect your team from electric spam, which would be devastating even with Ferrothorn around. Lastly, Tornadus is to kill Rillaboom. Any rain team without a hurricane user is an auto lose to Rillaboom. You can't go building a rain team without a hurricane user because are you really that willing to bet your odds that you are never gonna run into one?
Sun teams are also the same. The standard sun teams usually consist of Torkoal, Venusaur, Heatran and Mandibuzz accompanied by whatever else. Why is this? Because if you are not using Venusaur, don't use sun, at all, and without Heatran, you could accidentally end up powerup up opposing fire type moves and without Mandibuzz, sun teams get destroyed by ghost types and sd Garchomp
You see a pattern? When using weather teams, you are not just building one for the fun of it. You have to follow a specific structure otherwise your weather team will be glaringly flawed. Of course, you can use different mons from the one I mentioned in their respective weather but those mons you use have to be able to fulfill the near exact role as the ones I mentioned. Naturally you can mix up the roles and sometimes even skip one entirely but if you are to do that then you really need to be experienced with that playstyle and how you deal with threats
While that only applies to automatic weather, manual weather is a different story and this would actually depend based on the generation. An example of this is swift swim spam in gen four uu. Because only King Kyogre had drizzle at the time, you had to use rain dance if you wanna use a weather team. However, based on FSG's videos on youtube, it would all depend on how viable the weather abusers are in the tier they are in. If rain is more viable, then it will be used more. The same is also true for sun. The reason why multi weather teams are only run by noobs, aside from the reason I mentioned earlier, is that you can only fit so many mons on a team. Again, you're weather team still needs to be defensively safe somewhat and it needs to be offensively threatening and you cannot put more than one weather to meet those two requirements, automatic weather or not
So, are multi weather teams viable? Absolutely not if your goal is to win. If not, then enjoy losing three hundred elo in an hour