I've used a few different cameras over the years, to video gymnastics,. My opinion is you pay a premium for "camcorder" size and convenience, when compared to image quality. Sorry if this gets a little technical, but since you're a DSLR user and have a 70-200mm f2.8, you probably understand the basics of aperture and sensor size, etc.
I started out taking video with a DSLR and similar lens, which worked well, but I quickly decided that I wanted to take both stills and video.
I purchased a Panasonic FZ1000, which has a 1 inch sensor, 25-400mm f2.8-4 lens (8x zoom) and 4k. It can also do 120fps slow motion. It worked very well at 30fps 1080p. I never did too much with 4k, as I didn't like dealing with the huge files and I felt I was sacrificing some low light performance for higher resolution. These can be had for $300-400, which is a lot less than camcorders that use a similar 1 inch sensor. It is still semi large, but much smaller than you DSLR lens combo. I also felt when I started going to 60fps and 120fps, things got fairly grainy/noisey, particularly in poorly lit gyms.
The next camera that I purchased last year is a Canon M50. This is a very compact mirrorless camera, that has a DSLR size APS-C sensor. With the EF-m 55-200mm lens, it is still very compact. Even though the apeture is 4.5-6.3, the larger sensor size makes up for the smaller apeture. I am able to do pretty clean 60fps video at 1080p, which gives smoother motion for action. It also allows for smother slow-motion. Though it will do 4k, I value the higher frames per second over resolution. This combo is going to be a in the $800 range new.
Again sorry if this is technical. I have unlisted YouTube videos of my daughter's meets. If you message me I can send links to samples from both cameras.