I have both 70-200/2.8 and the 100-400 II lenses (also 300/2.8, 300/4 and 70-200/4)... and I use them a lot for equestrian photography, mostly on a pair of 7DII cameras that have essentially the same sensor as your 70D.
The 100-400 II is and excellent outdoor lens, but just not "bright enough" for a lot of indoor shooting. With f/5 up to about 300mm and f/5.6 beyond that, I cannot use it in many covered arenas without having to use slightly too slow a shutter speed or a little bit too high an ISO (and I use up to 6400 pretty freely with my 7DIIs). Indoors I usually use at least f/4 lenses... or, with less high ISO-capable cameras in the past, f/2.8 lenses. I also find that I rarely need more than 300mm on an APS-C cameras, with typical arenas. But I am the "official" photographer at the events and might have closer access than you usually enjoy (during some of the more sedate events such equitation and pleasure "rail" classes, I'm often in the arena itself, with the competitors... though in those cases I often switch to 70-200mm and 24-70mm lenses on my cameras).
Where I use the 100-400mm most is outdoors, with larger arena events such as gymkhana. It's very sharp and quite fast focusing, plus has the latest-and-greatest, high performance image stabilization. It also would be excellent for wildlife photography... probably better than any shorter lens or zoom combined with a teleconverter. (On your 70D, the 100-400mm is not usable with 1.4X... on more recent models such as 7DII and 80D, it is.)
Now, the 100-400mm II is fairly large and heavy, too (about 3.5 lb.) I use it handheld a lot, but for longer sessions I'll usually put it on a tripod (Gitzo series 3 with a gimbal adapter). It's NOT an "Internal Focusing/Internal Zooming" lens ("IF/IZ", the way all the Canon 70-200s are), so it grows longer, which upsets balance slightly on a gimbal.... But that's manageable. It DOES have sophisticated IS that doesn't need to be turned off when using it on a tripod (the first version 100-400 didn't.... neither does the EF 300/4L IS.... their IS should be manually turned off if using them locked down on a tripod.)
One thing that's important if wanting to to use the 100-400 II on a tripod or monopod.... the OEM tripod mounting foot is "stylish and conveniently removeable", but simply doesn't work well with Arca-style quick release lens plates. Fortunately though, there are replacement tripod mounting ring feet being offered by RRS, Kirk, and Hejnar Photo. I ended up buying and use the Hejnar.
I have not used a 2X with 70-200/2.8 II. I would strongly encourage you to research that combo very carefully before going that route... to see if the resulting image quality will be acceptable for your purposes. The "II" version of the lens and the Canon 2X "III" are said to work together very well (
much better than any of the other Canon 70-200/2.8s with any of the 2X Extenders). But there's inevitably some loss of image quality, as well as some slowing of autofocus performance. Many people find it acceptable... but some others don't. Maybe a 1.4X would give you "enough reach" for your purposes.... there's a lot less loss of IQ with the weaker TCs (not to mention, a stop less light lost). I frequently use a Canon 1.4X II on 300mm and 500mm primes (and occasionally on 135/2). But I haven't needed to use it on any of my zooms. It works very well with 300/2.8L IS, 500/4L IS and 300/4L IS. For example, below was shot with the 300/4 + 1.4X combo, here on 5D Mark II...
And the same combo on original 7D...
I also have Canon 2X II Extender, but use it pretty sparingly and only on my 300/2.8L or 500/4L primes.
I also mentioned the EF 70-200mm f/4L IS USM that I use. It's about 2/3 the size and weight of the 70-200/2.8 lenses. I originally bought it as a backup, since 70-200 are one of my most used lenses. But I now find I actually use it more often, since I ain't gettin' any younger and have really learned to appreciate the lighter weight and smaller size. The 70-200/4 is also quite high performance with excellent image quality...
The 100-400mm and most of the Canon 70-200mm lenses (except for the oldest f/2.8s... the one without IS and the original IS) use fluorite elements that contribute to their high image quality. Canon has used fluorite in many of their telephotos for decades, even in their FL/FD series prior to the modern EF/EOS lenses. Nikon has recently converted many of their telephotos to use fluorite, too (including their latest 70-200mm f/2.8). But AFAIK, Canon and Nikon are the only lens manufacturers using it. Fluorite is particularly effective counteracting chromatic aberrations, among other things. But it's rarely found naturally occurring large enough for lens elements, and it's difficult to work with. As a result, it has historically been pricey. For example the Nikon 70-200mm FL costs $2800, which is about 33% more expensive than their prior version without it). However, Canon pioneered both growing their own fluorite crystals and methods of working with it, so has been able to offer lenses using FL that cost as little as $650 (not to mention that Nikon's 70-200 FL costs almost 50% more than Canon's 70-200/2.8 II).
All the above are USM lenses... And that - especially along with f/2.8 lenses (without any teleconverter) - enables the highest performance AF when shooting stills of fast action. The 300 f/4 and 100-400mm II are ever so slightly slower focusing than f/2.8 lenses... hardly noticeable in reasonably good light. Overall, a camera upgrade wouldn't do much for you... EXCEPT in this respect. Your 70D's AF is probably rated to -1 EV. Next generation models such as 7D Mark II and 80D are now using AF systems rated to -3EV, so might be a little better performing indoors in more challenging lighting. ALSO, I
think your 70D has the "Flicker Free" feature that's very helpful under a lot of types of indoor lighting... but if it doesn't, both 80D and 7DII have that feature. It makes a huge difference in the percentage of accurate exposures under sodium vapor, fluorescent and some other types of lights. But, to be honest, I used a pair of original 7D for five years without any problem, including a lot of indoor/covered arena work. Your 70D has essentially the same 19-point AF system as those cameras used.
Finally, you also mention shooting video and may want to switch back to your STM lenses for that. Most USM lenses are not ideal for video work, if you are using AF and/or recording sound. According to Canon, USM is 2X to 4X faster focusing than STM (using their EF-S 18-135mm example, because that one of the few lenses, maybe the only one that's made with both types of AF drive).... BUT, STM is smoother and quieter, so may be preferred for video work. Note: Several of the latest Canon cameras are using a new form of "Nano USM" that's both fast for action shooting AND good for video work. However, AFAIK so far this includes only the EF-S 18-135mm USM, EF 24-105mm f/4L "II", and EF 70-300mm IS USM "II" are utilizing this new type of focus drive.
Hope this helps!
I have both 70-200/2.8 and the 100-400 II lenses (... (