I'm not really sure about twitter specifically; I don`t use it. But when researching POTUS' tweets I came across a couple of good websites and figured that they are using twitter's api.
For people that don't program, an "api" is an "application programming interface," which is basically a set of tools for you to get what you want, designed by the maker of the app. It works in the best interest of these social media companies to develop a good api because it allows others to use and display their stuff on other websites–free advertising and spread of influence. I'll look further into it, as it relates to an app I'm working on.
As far as using HTTrack on it, I haven't tried. For the most part I've had success with websites that don't involve user accounts; there may be a way around that, but again, I'm pretty new to it myself. Just taught myself like 4 days before Q highlighted liddlekidz.