What they mean by independently is that CoG won’t do anything other than publishing. They won’t provide editing or make any adjustments to your story. So you’re going to code, write, and proofread your story by yourself. Yes you can work with a yeam, but not with CoG.
You can run a beta by using a site like dashingdon (which is popular for hosting betas for games). You just simply have to create a topic in this forum underneath the Works in Progress category with a name like STORY NAME-WIP BETA and so on (make sure your topic includes the dashingdon link). I recommend adding tags to reach your target audience. These betas are important since they allow your game to gain traction and a following, receive feedback and bug reports. This is especially important with HG since unlike CoGs, there is no quality control.
Once you’ve got that down, the Hosted Games publishing area of this website states that you need to email the game’s txt files (the scenes, etc.) for reviewing (again, they will not proofread or correct bugs, they will just review the content of the game). For more info, go to https://www.choiceofgames.com/make-your-own-games/hosted-games-submission-form/