Without a voting system, users have no way to find "quality" content. Everything is mixed together, which means that the signal to noise ratio is only going to get lower and lower as more people contribute to the community.
The voting system is extremely important if not necessary for any sort of forum site. Like phail said you won't be able to find quality content, which means if the site takes off or gains enough followings it will devolve into chaos of shitposting, fake news, and garbage. Of course there still will be good content yes but you won't be able to tell one from another.
I consider that to be a feature. Voting systems mean that homogeneity is prized first and foremost, since most people only look at what's highly voted. This leads to a situation where the site will foster users with particular views and biases. Voting systems can also be gamed (like on Reddit), when you can pay some people to bot-upvote your post. Applied to comments, it's even worse. "Echochambers" on places like Reddit aren't formed just from mod bans, but because they attract users who upvote content that appeals to their biases and downvote the rest. That's how they control the discourse.
This is in my view one thing 2channel-style boards get right. Not everything has to be about "quality content", and it does a bit of a disservice to users to say that they have such little control that they can't tell good content from bad. Humans aren't animals, we don't just take what we're given. Sites like Reddit, HN and Twitter destroy that agency within us, and it's a shame.
Here is an essay by Shii, one of the most well known historians of imageboard culture on the value of anonymity: http://wakaba.c3.cx/shii/
While I agree with you that voting is required for filtering, voting alone is not the best way to find engaging content in a discussion focused website and so I think that a post with most comments or replies is what makes it engaging and that's the focus of this website.
You can do without voting. All you need is a way to discern what is popular, in order to display 'popular' and 'new' categories. For example by measuring comments and clicks, maybe the amount of time users spend on the comment page relative to the amount of comments. Not a great idea since it can be gamed, but what can't nowadays.
It would probably favor clickbait, violence, etc like every other system. The only escape from this on a user-submitted site, if there are no subcategories outright banning certain content, is what's already in place, just sorting everything by date.
content is not valuable just because someone clicked it, you cant know its valuable until after having read it. otherwise its all about the headline, which is a bit of a reality of modern day news, but this would just make that situation worse.
secondly just because you dont want to comment doesn't mean its not valuable.
so basically this just seems like a plainly terrible idea.