Aaron, the process I use for taking longer streams of Tweets to own them (via PESOS) has Kevin Marks‘ excellent tool Noter Live at its core. Noter Live allows you to log in via Twitter and tweet(storm) from it directly. As its original intent was for live-tweeting at conferences and events, it has some useful built in tools for storing the names of multiple speakers (in advance, or even quickly on the fly) as well as auto-hashtagging your conversation. (I love it so much I took the time to write and contribute a user-manual.)
The best part is that it not only organically threads your tweets together into one continuing conversation, but it also gives you a modified output including the appropriate HTML and microformats classes so that you can cut and paste the entire thread and simply dump it into your favorite CMS and publish it as a standard blog post. (It also strips out the hashtags and repeated speaker references in a nice way.) With a small modification, you can also get your site to add hovercards to your post as well. I’ll also note in passing that it’s also been recently updated to support the longer 280 characters too.
The canonical version I use as an example of what this all looks like is this post: Notes from Day 1 of Dodging the Memory Hole: Saving Online News | Thursday, October 13, 2016.
Another shorter tweetstorm which also has u-syndication links for all of the individual tweets can be found at Indieweb and Education Tweetstorm. This one has the benefit of pulling in all the resultant conversations around my tweetstorm with backfeed from Brid.gy, though they’re not necessarily threaded properly in the comments the way I would ultimately like. As you mention in the last paragraph that having the links to the syndicated copies would be useful, I’ll note that I’ve already submitted it as an issue to Noter Live’s GitHub repo. In some sense, the entire Twitter thread is connected, so having the original tweet URL gives you most of the context, though it isn’t enough for all of the back feed by common methods (Webmentions+Brid.gy) presently.
I’ll also note that I’ve recently heard from a reputable source about a WordPress specific tool called Publishiza that may be useful in this way, but I’ve not had the chance to play with it yet myself.
Clearly, you can embed Tweets, often by adding the URL. However, there are more and more people deleting their Tweets and if you embed something that is deleted, this content is then lost. (Not sure where this leaves Storify etc.)
It’s interesting that you ask where this leaves Storify, because literally as I was reading your piece, I got a pop-up notification announcing that Storify was going to be shut down altogether!! (It sounds to me like you may have been unaware when you wrote your note. So Storify and those using it are in more dire circumstances than you had imagined.)
Storify announces it will disappear from the web on May 16, 2018. Once a core part of social-focused journalism projects like @acarvin‘s work, it’s larger archives and URLs will be gone. https://t.co/9KhEYCbX2e
— Aram Zucker-Scharff (@Chronotope) December 12, 2017
It’s yet another reason in a very long list why one needs to have and own their own digital presence.
As for people deleting their tweets, I’ll note that by doing a full embed (instead of just using a URL) from Twitter to WordPress (or using Noter Live), that the original text is preserved so that even if the original is deleted, a full archival copy of the original still exists.
Also somewhat related in flavor for the mechanism you’re discussing, I also often use Hypothesis to comment on, highlight, and annotate on web pages for academic/research uses. To save these annotations, I’ll add hashtags to the annotations within Hypothesis and then use Kris Shaffer’s excellent Hypothesis Aggregator plugin to parse the data and pull it in the specific parts I want. Though here again, either Hypothesis as a service or the plugin itself may ultimately fail, so I will copy/paste the raw HTML from its output to post onto my site for future safekeeping. In some sense I’m using the plugin as a simple tool to make the transcription and data transport much easier/quicker.
I hope these tips make it easier for you and others to better collect your content and display it for later consumption and archival use.