Found embetty, and now I am reconsidering to embed marked-up Tweets and YouTube videos again. Embetty displays Tweets as proper cards, but without Twitters visitor-tracking code.
Instead of screenshots, they have a demo page.
You need to setup an embetty-server first, though. That's kind of annoying, but not much of a problem for pros. I wonder if a shared server would do, or if that puts you into weird legal situations again, because the embetty-server proxy now becomes the de facto hub of displayed content but isn't allowed to according to Twitter and YouTube rules, or something like that.
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