Answer

This question is from James Thurber's The Catbird Seat. The original text is as follows:

• There was nobody on the stair, which went up ahead of him ... A door opened down the hall in the wall on the right. He went toward it swiftly, on tiptoe. "Well, for God's sake, look who's here!" bawled Mrs. Barrows, and her braying laugh rang out like the report of a shotgun. He rushed past her like a football tackle, bumping her. "Hey, quit shoving!" she said, closing the door behind them. They were in her living room, which seemed to Mr. Martin to be lighted by a hundred lamps. "What's after you?" she said. "You're as jumpy as a goat."

This clearly shows that he did not injure her in any way, but merely surprised her and shoved her. The following is from Critical Semiotics, Lecture Eight: A Semiotic Reading of James Thurber's "The Catbird Seat" by Scott Simpkins¹

• When Mrs. Barrows opened her door... she appears as a formidable barrier...Mrs. Barrows would normally stand as a considerable annoyance for Mr. Martin... But, newly invigorated, Mr. Martin "rushed past her like a football tackle, bumping her."

This is a tricky sentence to interpret because there are two meanings of "a football tackle: a verb and a noun." (AmEng). In addition, your reference to "swift move" and "football tackles on YouTube" got pretty much everyone² thinking in terms of the verb. (A verb-ish meaning could make sense, even though it would be awkward. It would be similar to "a train wreck" which is a noun form of "The train did wreck." This is sometimes used like this, "The movie was painful to watch; it was like watching a train wreck in slow motion. You knew it was going to end badly.") Here are two definitions for "football tackle".

• football tackle (Verb): The action of one person knocking down another person who is holding the football. "He tackled the quaterback."
• football tackle (Noun): A person designated and positioned on the field to attempt to tackle the person who is holding the ball. "After graduating in 1941, Clarence attended Morgan State University in Maryland and twice earned All-America honors as a football tackle." (African-American Sports Greats: A Biographical Dictionary, David L. Porter)

The proper interpretation is the noun form. The sense of the sentence is like, "John ran like a cheeta." Also note that a "rush" in football is a play in which several tackles will rush (run quickly) towards the quarterback to grab him and take him to the ground (tackle him).

The word "like" is important, since that suggest an simile, wherein one thing is similar, but not the exact same as, another thing. If one says, "My brother ran like a cheeta" we don't think he really ran as fast as a cheeta. We know that means he ran very fast. So "like" describes how he rushed: like a tackle might rush towards a quarter-back. He bumped her as he passed her. It's an aggressive and surprising movement for sure. However, it was certainly not as fast or aggressive as a football player. And importantly, he did not tackle her or even attempt to tackle her! That character had enough of being intimidated by her, and the bump was to show her that he wasn't going to put up with her domination anymore.

<sup>1. A "semiotic reading" would (roughly) be one that analyses what is meant, symbolically.
2. @snailplane determined correct interpretation. @J.R. helped bring to my attention with co-editing.</sup>

Additional "Advice"

I would like to offer some additional advice, since I've noticed that you're asking quite a few questions from famous short-stories that may be somewhat advanced for you. It's difficult to give you a good sense of what's going on without more context, and typically these books are studied in class situations where themes play an important role in understanding.

The details of what these "mean" both literally and figuratively have been covered by analyses that are available on the web, and you might want to start there first. Also, you may find famous stories such as this one (and commentaries about them) have been translated into (or written in) your native language as well.

Yes, they have different meanings. In the context of Fifty Shades, I don't do romance means the character is wholly uninterested in the usual trappings of attracting a mate: wining and dining, emotional intimacy, evoking tender emotions, etc. The speaker is solely interested in sexual activity. On the other hand, someone who says I'm not romantic might be interested those things but not very skilled at them.

I don't do X means I'm unwilling to engage in X. It connotes unwillingness and does not generally convey inability. See the extremely topical question "What does 'I don't do dogs' mean?" for an in-depth exploration of this phrase. This is a modern construction, so it's unsurprising that you didn't find any hits in your N-gram search.

I'm not romantic expresses a more fundamental statement about the person's nature. Romance is alien to the speaker's personality and emotions. It would be difficult, and perhaps impossible, for them to experience or express it.

There's also a subtle difference between romance and romantic. In this case, romance is an activity, while romantic is a descriptive quality. An unromantic person might well be involved in some romancing, but it would likely either be awkward for them to attempt (because romance is not part of their nature), or one sided (someone else attempts to romance them, but encounters difficulty, because they don't experience romantic emotions much or at all).

Conversely, someone who doesn't do romance may be fully capable of experiencing romantic feelings, but they would surely stifle and not express them. Likewise, they would categorically reject all attempts to romance them, whereas someone simply unromantic might warm to the idea, or at least appreciate it.