If reach or budget is important, 3rd party lenses are tough to beat, like the Sigma 150-600.
If AF speed and IQ (sometimes, depends on lens) is important, Canon OE lenses are tough to beat.
That being said, Sigma lenses for example have good AF tracking speeds, it is their initial acquisition speeds that are terrible. That impacts a shooter in two areas:
- If the lens focuses off the subject and goes to the background or foreground, it can be very difficult or impossible to get back to the subject before the moment is lost.
- Seeing a subject and getting the system to latch onto it on time to get the shot.
Once it has a lock, and you can keep the lock on the subject, movements of the subject likely won't tax the AF system of a 3rd party lens, the newest offerings from Sigma, etc are quite good at AF changes on a locked subject.
I know the topic is only covering 70-300 or 100-400 variants, but keep in mind that to get the reach of a 400mm lens on a 300mm, you will need to add a teleconverter, which usually comes with a bit of baggage in the way of light transmission, AF speed, etc. This is where maybe looking at a 3rd party really long lens like a Sigma 150-600 makes sense. 600mm with a 70-300 and 2x vs a native 3rd party lens at 600mm will likely suffer IQ-wise, I would think.