I think this is a difference between the desktop and web experience because I don't see any "more tweets" section. If I scroll, I just get autoloading of more replies to the tweet.
Not sure whether you’re on desktop or mobile, but I only get “More Tweets” on the mobile website if I’m not signed in. Otherwise it’s as you describe, only replies.
I have made more observations and it seems to have to do with how you arrived at the post. If I click on it from browsing my timeline, it does not show “more tweets“. If I click through to it from a link somewhere external, such as a link in an article, it does. I’m sure that somewhere there is a thorough analysis of this.