I think that wouldn't work because of copyright and Terms of Use. Probably somewhere in Twitter's ToU there's something that says you cannot copy Twitter discussions and add to your own website. — If you study Twitter's ToU, probably the people who post to Twitter, never gave you permission to include their tweets on your blog.
I like the idea, and would want to do sth simiar, but just linking to tweets, instead of "downloading" them to the blog. And the blog author could write a summary of what the Twitter discussion is about (what sub topic related to the blog post, the Twitter discussion discusses), next to the link.
I'd also like to automatically find and link to discussions at HN and Reddit and Mastodon (in a commenting system I'm developing, seem my profile)
Here an example:
https://nicolas-hoizey.com/2017/07/so-long-disqus-hello-webm...
The missing part is the creation of the initial tweet in your idea, but that should be easy to automate.