Well, I have both Canon whites and a Tamron 150-600. The differences are pretty stark honestly. I like both for what they do. The 150-600's really are excellent for their cost and they are awesome for stationary birds and will do a very good job, even compared to a big white combination. But that's stationary birds. You can get away with in-flight shots with the big 150-600's, but they will not be as fast (both AF and focal-ratio) as a big white, especially being driven by a lesser AF system. I've used my 150-600 for in-flight a lot, and while it's doable, it's definitely not its ideal use. It's excellent for the stationary stuff at distance or simply small things. But, I use a 300 F4L IS for BIF for a reason these days, it's simply blazing fast, crazy fast, on my old 7D. Not even a big 500 F4 on a full frame, but a very similar FOV and different pixel density. Regardless, I love the 7D + 300 F4 setup because it's very light weight, yet has aggressive fast AF and a blazing fast AF lens and gobbles up light. I was about to pull the trigger on a 400 F2.8L, but the 13lbs really put me off, I knew I'd never use that for much without a tripod/monopod. So then I almost got the 300 F2.8L because it was lighter and I could use it with TC's still, but the weight was still a put off. So I settled on a 300 F4L IS because it's sharp and fast. I'm often shooting from a kayak, and in the swamps, and as much as I'd love that beast, or a 500 F4, I know that those are not realistic for my purposes when I'm really close to the subject, and need really fast, light weight performance. So, the 7D + 300 F4 is what I use. I still use my 150-600, but I rarely use it for in-flight. It's more of something where I just need the reach and have time to set it all up. For the spontaneous raw speed of something up close, that's what a white lens is for.
The camera matters too. Using a very good aggressive modern AF system, like Canon's, is a world of difference from something else when you're tracking a fast moving bird at close range. Canon still pretty much leads the pack when it comes to aggressive excellent AF and has some of the best big white lenses to match. There's a reason you don't need certain bodies out there and certain lens makers out there, lined up, shooting fast moving raptors in flight taking down prey. So this is why people spend thousands on one of those setups.
But, if you want to shoot stationary wildlife/birds and some large birds in flight at distance, most any modern camera and a big 150-600 will work out just fine actually.