The reality is that relying on a fair use defense for this is risky in the U.S.
Here are some of the major issues involved.
Is the broadcast copyrightable?
It's fairly well settled in the U.S. that when a public performance is simultaneously broadcast, that broadcast constitutes fixation in a tangible medium of expression under 17 U.S.C. § 101. The broadcast itself is then protectable.
Side note: whether a bootleg video shot by a fan in the stadium should be protectable under federal copyright law is hotly debated. There are good arguments to be made on both sides. The matter of the only surviving copy of Superbowl I is a good example. That said, making personal videos of the games is often a violation of the fan's ticket contract or a state-level law.
What rights does the broadcast owner get?
Among other things, the owner gets the exclusive right to reproduce it, to distribute it, and to prepare derivative works (like highlight films). Posting the owner's broadcast on something like Instagram or YouTube is likely an infringement of these rights unless a defense like fair use applies.
Is this a fair use?
The trouble with fair use is it's a highly fact-intensive inquiry, and courts aren't consistent in the way they apply the doctrine. The basic idea is that 17 U.S.C. § 107 has four factors the courts are supposed to look at when they decide whether something is a fair use. For example, here's a fair use analysis related to email attachments; here's a more general discussion of the factors.
Factor two (the nature of the copyrighted work) will probably go against you: audiovisual works like broadcasts tend to be clearly protectable. However, factors one, three, and four leave more room for debate.
In your case, you've brought up the third factor: the amount and substantiality used in relation to the broadcast as a whole). Your argument would be that taking 20 seconds of a game is small relative to the overall length of the broadcast. The broadcast owner would be able to make a strong argument that while you have taken a small amount, the clip is nevertheless substantial since goals are the "heart" of the broadcast. Judges will come out differently on this factor.
You'd argue that factor one (purpose and character of use) goes in your favor since you're not distributing the clips for commercial gain. That does help your case. However, the broadcaster would cite any number of opinions that downplay the significance of a commercial purpose and instead focus on whether your usage some how "transforms" the broadcast thereby providing something new to the world. Unedited clips of goals are probably not transformative, so you'd have to hope you end up in a court where the judge doesn't care about the transformation argument (which is hard to do since the broadcaster gets to choose where to file the suit).
Some courts think of the fourth factor (harm to the copyrighted work's market) as being the most important. Here the relevant inquiry is whether your posts harm the owner's market for derivative works (which are a right we discussed above) like highlight films. The analysis might go something like this: sharing highlights on Instagram means some people who might otherwise buy the owner's highlight films aren't going to do it anymore--and that's a harm. So the analysis would have to go on to see how significant that harm would be, which is where the lawyers would introduce evidence and expert testimony.