I think it's so awesome that you are interested in books. For me, I'd recommend goodreads.com if you're not a part of it already. For me, I've found it a source of inspiration to connect with friends which help me find books to read and enjoy. There's also a community portion of goodreads, which would fulfill the book club aspect that you're searching for.
I'm not familiar with any websites that read to children (except for the few I found via google search after I saw your question). But have you thought about starting your own youtube channel? Your question reminded me of this video that I saw a few months ago, which was one of the best things I saw on youtube, that I promptly shared w/my kids as they love Dr Seuss: https://www.youtube.com/watch?v=hqIbEHNqbPs .
Anyways, I wish you the best of luck! If you need a good reads friend to start things off, feel free to add me: https://www.goodreads.com/user/show/49348142-dexter :D
When I was in high school, I volunteered with SMART (Start Making A Reader Today) where we read to first and second graders, so I know what joy it is to sit and read with a child. Reading with my nieces through FaceTime isn't the same as being there in person, but I still enjoy it. :)