Late reply, but it got me thinking.
It seems like there are two lens paths for zooms for e-mount.
The first contains the 24-70, a 70-200, and either a teleconverter or the 200-600 if you need the extra reach.
The second contains the 24-105 and the 100-400.
Personally, I was considering the second route until I read your post and really thought about it. In my APS-C days my most used lens was a 70-300 that saw a lot of use on the long end of its focal length at the side of racetracks. I've since had kids, and COVID happened, so I haven't been to a racetrack in years. I'm hoping that will change in the next year or two now that my kids are older. I probably won't be going out 3-4 times a year though. A more compact 70-200 with a teleconverter could be a good choice. If I were to go this route, I might favor the 2.8 some due to the 2 stops lost with a teleconverter. I do worry that 70mm might be too long on the short end for dad duty though. The 24-105 is a good "I'm going to be wandering around shooting things kind of close to me" lens and the 100-400 is a good "I'm going to be wandering around shooting things far away from me" lens. 70mm is pretty tight and 200mm doesn't give you that much reach. Granted, if you're going the teleconverter route this could be a good "shooting things far away" choice.
What's your use case? If you're going to be shooting in a well lit environment without a teleconverter the f/4 lens is where I would personally land. With that kind of focal length you'll easily be able to blow out the background if you want to, even at f/4. If you're shooting in true low light you're probably going to want an even faster lens than f/2.8, but at these focal lengths that can be a challenge. What other lenses do you own? Does this focal length overlap or complement your current lenses?