I shoot with Canon gear, not Nikon... but among my most used lenses for sports (incl. a lot of equestrian) are 70-200mm.
I have two of them.... an f2.8 with image stabilization (about 15 years) and an f4 with image stabilization (bought more recently). A couple years ago I bought the f4 lens as a backup, since this has been one of my most frequently used lenses. But I find I now use it more often than the f2.8. It's approx. 1/3 smaller, lighter and I was rarely shooting with the f2.8 lens wide open, anyway, so the f4 is no problem. I mostly only get the f2.8 out for low light shooting now... and I suppose it serves as a backup to the f4 lens!
Two things:
1. 200mm really isn't long enough a lot of the time for sports, even on the APS-C (i.e., DX) cameras I use. Especially since you are using a full frame FX camera, I hope you have a 300mm or 400mm in your kit, too! I use the 70-200mm for the closer sports action and a second camera with either a 300mm prime (sometimes with 1.4X teleconverter added) or a 100-400mm zoom on it, switching to them for the more distant action.
2. With both lenses, as well four or five others with it, I leave image stabilization on all the time. With the 70-200s I do regardless whether I'm hand holding, on a monopod or on a tripod. However, Canon's IS is almost certainly different from Nikon's VR (must be, since they both use patented technologies). Many Nikon users feel that VR slows down auto focus and there's some anecdotal evidence to support that. After using them for about 15 years, I feel Canon's IS does the opposite... If anything it seems to help auto focus speed of acquisition and tracking ability. Also, while a few Canon lenses require IS to be turned off when on tripod, that's only when it's fully locked down and there's absolutely no physical movement of the lens... certainly not the way I'm using a lens on a tripod with sports. Plus, many Canon lenses, including all their 70-200s with it, sense lack of movement and turn off IS automatically. I figure, if nothing else, IS or VR can be turned off if you don't need it.... turned back on if and when you do. (Canon still sells two versions of 70-200 with IS and two versions without it. Primarily for the lower cost of the lenses, some sports shooters opt for the non-IS versions and are simply careful to keep their shutter speeds fast enough.)
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As someone else mentioned above, VR does nothing for the photographer when shooting action. Most sports photogs (me included) turn it off when on the sidelines.
That maybe true of Nikon users. But it's not the case with Canon shooters. Sure, some turn IS off... Mostly to save a little battery power. But a lot of Canon users leave it on, too. After all... at least with Canon's IS... it can't hurt, might even help. Heck, I've got the switch on several of my lenses covered with gaffer tape, to be sure I don't accidentally turn off IS!
As to the weather sealing... Personally I think that's not much of a consideration, for one reason: If I'm out shooting in nasty weather, I'll protect my camera and lenses anyway. I'll use a rain sleeve, plastic poncho... heck a plastic baq from a grocery store or a shower cap from the hotel room have served at times. Truth is, none of these cameras are entirely weather
proof. At best they are weather-
resistant. About half my lenses, including both 70-200s, are supposed to have more advanced sealing... but I don't treat them any differently than the ones without. I take extra precautions with all of them, as much as possible.
Much as I try to avoid it, I have gotten caught out in nasty downpours a couple times with both weather-resistant and less-well-sealed cameras, lenses and even flashes. IN the worst soakings, I simply protected it as best I could and immediately turned it off, removed the batteries and dried it thoroughly for 2 or 3 days before powering it back up again. Never had a problem or failure with a modern camera, lens or flash, due to weather-related situations.
I don't know about they models you asked about, but Nikon's new 70-200/2.8 "FL" is supposed to have incredible image quality. AFAIK, it's the first of Nikkor zooms to use fluorite. Most of the Canon have used FL all along... even the lowest priced one (non-IS, f4... about $700).