The feature you're referring to in the first link you provided in your question is available only on Twitter mobile. To add or thread a tweet to an old tweet all you need to do is to reply it. Technically speaking, the feature mentioned by Twitter is simply bringing an old tweet to view based on context while re-packaging the reply button. On a desktop you can do it yourself by finding the old tweet you wish to thread a new one below it, and then simply click the reply button on it to attach a new tweet to it.
Even if your old tweet had already accumulated lots of replies, if it is you who reply to your own old tweet, Twitter will thread your new reply tweet at the bottom of the old tweet and not in its chronological reply order.