I'm in Atlanta and I do it all the time. I put a GoPro on the fence behind the backstop to record the full game as my son is a catcher and I want to capture what he does as a catcher, not just as a hitter. Other parents really appreciate the game video and highlight clips I am able to make for them if their kids made a good defensive play. Most parents take videos of their kids' at bats. But it's hard to capture a good defensive play unless you take a video of the full game. I also reach out to parents of the opposing them if they made a really good play and I offer to send them a highlight video of that play. I made lots of good friends from other schools that way.
Livestreaming is a bit more challenging due to getting a connection to the Internet. I can't connect my GoPro to our school's wifi for home games. It's even worse for away games. I've used phones as hotspot for the GoPro but the speed is hit or miss. It's also easy to use up the hotspot data cap. If the parents really want me to try to stream for their grandparents or parents who can't make it to the game, I ask for volunteers to let me use their phones as hotspot. I was able to stream maybe 50% of the time last year.
I never had issues with school or state varsity rules not allowing me to stream.