Is Tamron 70-300mm an upgrade from Sony 55-210mm?

[jncyphers]

Hello.

I have Sony a6400, a6100, and ZV-E10 cameras. I have been enjoying photography as a hobby for a couple of years now. I normally photograph wildlife. My 8yo son is playing football, and I volunteered to take pictures at the first game. My sister took some pictures while I was tasked with something else at one game. The head coach was very happy with them and politely asked us to continue to take pictures at each game. I'm happy to do it, but I only have one telephoto lens. I'm very willing to buy a new lens, but I don't want to spend a fortune on a second zoom lens. I have a Tamron 28-75mm lens, but we found that 75mm isn't enough for close shots.

We are going to use the ZV-E10 for videoing the game from the stand. The camera we used for game one was the a6400 with a Tamron 150-500mm lens. We were very happy with the results. However, that's a $1,200 lens; I would like to spend less this time. I have a Sony 55-210mm lens, but I was hoping to use that on the ZV-E10 and buy a lens a bit nicer than that one for the a6100 for photos on the other side of the field from my sister with my a6400. I was considering the Tamron 70-300mm lens, but is this lens really any better than the Sony 55-210mm that I already have? Is there something around this price range ($400-$600) that would be better for in-game football pictures from the sidelines? If there isn't anything in that price range, how much would I need to spend to upgrade from the Sony lens?

Thank you!