No, not at all. Because banning Swift Swim will completely and almost utterly make rain useless as compared to banning Drizzle.I don't see why you couldn't tech Rain Dance as a move, I mean, people are using things like HAIL on Tentacruel, Vaporeon, Burungeru, and other weird things to stop Rain/Sun/Sand, it is far easier to throw Rain Dance on and it has so many users that could set up at least a temporary rain that actually can make use of it. And Sand has LANDLOS, DORYUZU, AND GARCHOMP. How would that at all make Rain equivalent? Rain would lose all it's best Pokemon (Manaphy is probably going as well since it is broken with or without permanent rain, just less so without the permanent). People don't use Rain to abuse Thunder or Gale, it's just a bonus you can and some Rain Set uppers get stab as well on those (Tornelos, Zapdos, etc). Those are mainly for other Swift Swimmers and water pokes that wall your super stab water moves. Without Swift Swim, you take away the utility of Kingdra, Kabutops, Ludicolo, and other pokemon who would never see the light of Day in OU (or even UU some of them) without Swift Swim. They are not broken of themselves without Rain and permanent Rain is what made them broken, not Swift Swim. Swift Swim is what makes them good and purposeful, permanent rain is what breaks them.
The other pokes that could use it...don't need the weather to be good. Starmie, Kerudio, Zapdos don't NEED weather to be good. It'll make them even more insanely powerful (especially Kerudio) but they can still be potentially outrun and you wouldn't use a spot for Politoad (despite the awesomeness of double stab Hydro Pump) just for them.
Swift Swim has never been the problem, it was just a very good ability. It's good enough to had been used previously to base teams around setting it up, but obviously nowhere near broken when it was on a time limited basis. And as people said, the difference is the time it takes to set up Rain Dance as compared to getting an AUTOMATIC PERMANENT (unless another weather changer/move user is used) rain. Rain Dance can be Taunted, Stalled out, etc. It was balanced in getting 8 turns as you had to then give up a useful item slot or have only a short 5 turn limit to use it (which is what keeps Trick Room and things down. Imagine the horror of an auto-permanent Trick Room that can't be cleared by weather. They'd have to make some sort of normalizing move and ability). And it "could" be stalled out with difficulty, but it was possible. Having to continuously set it up was another weakness. But Drizzle changed everything. Now you have to lose momentum carrying your own weather changing move (which you don't usually get any advantage of since most of the time it has to be Hail or Sand since you definitely don't want to give further advantage to Rain/Sun teams facing you) and are on the defensive and it comes down to killing the auto-weather and then using a move to change it. It pretty much forces either a weather team of your own or a complete overcentralization around weather. Whether (hah) weather is broken or not, it is very difficult and forces you to use weather or obscure things to even hope to counter.
And I am all for NORMAL weather teams. You know, the ones that actually have to use a weather move to set up. All auto-weather abilities should be banned. This would allow you to keep weather teams and normal teams and keeps things that aren't broken without weather and monsters in weather good, but not so clearly domineering and overpowering that people are seriously considering banning them to only Uber (not they can't be used there but they themselves without permanent rain are nowhere remotely close to uber unlike Darkrai, Shaymin-S, Deoxys, Inconsitent were last time) which is a disservice to them and tiering. It's not Kingdra, Kabutops, and Ludicolo who are broken. It's not even them with a limited Swift Swim that are so overpoweringly broken. Very strong and useful, yes, but not to the degree seen here. No, it is clear that it is the addition of Drizzle that changed everything. You'd also see even more variety of teams since you now need slightly more support but things like Venasaur and Kingdra and certain others would be worth setting up Sunny Day and Rain Dance for. Kingdra is notorious of double Dancing before and is amazing at it.
And it's good seeing Venasaur, Charizard, Kingdra, Ludicolo, Victreebell, Shiftry, etc in OU. Not in their permanent weather (they're so horrible then that things like Kingdra and Venasaur are being considered Uber and if Kingdra goes, there is no doubt Venasaur will as well as Venasaur is the sun's Kingdra but even arguably better) where they are broken. It actually would be fun and awesome with actual set up weather as they are still definitely threatening and good enough then but not to the point of "omg ban it!" which some people would rather do then the obvious problem of permanent weather. Banning speed abilities kills many useable pokes that are completely legit but aren't because of permanent weather. Without permanent weather, you could still have weather and fun strategies, and not have to run weather or tear your hair out trying to fight weather. Like Trick Room and other fun things but not to the point of seeing it EVERYWHERE.
Rain Stall is still viable without Drizzle. You'll just have to set up Rain Dance and why you would play that over Swift Swim I don't know. But it is possible and Swift Swim is definitely not the problem, Drizzle is. But banning Swift Swim (and other abilities by extension like Chlorophyll and Sand Throw because you have to because it is ridiculous and hypocritical to only target Swift Swim then) kills a usage of pokes and would make things that would have at least some purpose in OU useless. You kill FAR more useable pokes and strategies preserving a minor and lesser used (no matter how you feel about it) one. And it is still useable, just weakened without permanent weather while without Swift Swim/Chlorophyll/Sand Throw, there are many more pokemon that are not.
Let's be honest here. Tyranitar, Garchomp, Landlos etc don't need Sand. It just benefits and adds something extra. But they are ferociously strong with or without. Banning Sand Stream is probably the least necessary of auto-weather but Sand would be even more dominant as a weather without rain or sun and you might as well. Auto-Sand also is annoying to set up weather teams and is the real problem they were lesser used and cuts their utility. Hail as the only auto-weather might be insane (then again if Hail was the only weather, all the other set up weathers if they manage to sneak in a weather move could wreck it but...) with Kyurem and the like. And Drizzle can not leave without Drought as the only thing really holding Drought back from being as good if not better than Rain is the inducer is less useful than Drizzle's. Without Drizzle, Drought would take its place and be just as terrifying and maybe more so with Venasaur, Heatran, Shandera, Tangrowth, Victreebell, Charizard, Mebjuka, etc over Kingdra, Ludicolo, Kabutops. And a lot of those are just as strong if not stronger than many Swift Swimmers with more surviveability.
The game is FAR more diverse without auto-weather than with. Imagine it without Auto-Sand/Rain/Sun/Hail. You'd have way more options, Sand would no longer be a requirement and therefore not a repeat of Gen4. Tyranitar/Garchomp/Landlos/Gliscor/Terakion have uses far outside Sand teams and sand isn't a requirement for them although it could be useful to run Sandstorm (yay, it would be a viable move then without auto sand) to counter other set up weathers. But they are phenomenal pokemon with or without sand. Extremely speedy pokes will be within normal natural bounds, either requiring an Agility, Scarf, or speed boost to go outside and it is actually possibly to run other things. Weather will actually become even more viable considering the number of set up users that they'd never do with auto weather and will be an awesome but fair strategy, unlike the monstrousity it is now. Weather hower won't be required to win and it is far more open to running your favorite pokes since you don't have to worry about needing to have a weather counter or you lose, period. Teams would look less the same and it is just a more fun game. Automatic weather is not healthy for an open metagame (it's fine for Ubers since that is completely different and they compensate for having overbalances in everything that are far beyond the norm of OU or less pokes in stats, power, move options).