Tweeting and gloating about an undisclosed settlement is the whole point of bringing the lawsuits forward in the 1st place. It's a WH-backed assault on the "fake news"
He doesn't disclose the amount because he has other lawsuits. If you're trying to get $250m from one and you settled for $3 million from someone else, you don't want that company to know you settled for $3m or that's what they will offer.
Tweeting and gloating about a legal settlement is clearly part of a legal strategy to get the other outlets to settle. Not disclosing the amount is part of a strategy to get get other outlets to settle for as much as possible. They won't know if he settled for $250 million (unlikely) or just a few thousand (equally unlikely). Rumor has it that the settlement amount was in the seven figures, but that is just people on the internet guessing.
It's likely that not disclosing the sum was part of the settlement agreement and a condition by the plaintiff. I don't think it's a legal but rather a political strategy, giving the kid's backers ammunition against the "fake news" and air time on having "lost by settling". Settling for 5k doesn't have the same ring to supporters as having their imaginations run wild on speculation. The sum maybe 7 figures as you say, these newspapers have businessmen balancing the cost of a drawn out legal battle and possibility of ending in a courtroom where 40% of the jurors would have already made up their mind.
Well keep in mind that it is probably the publishers liability insurance company that insures the paper covering most of the settlement. And yes, I agree that the non-disclosure of the amount is probably a condition demanded by both the paper and plaintiff's lawyer as part of the settlement. For the paper is it probably better to settle quickly for some amount below where they are insured.... and just move on... and avoid a disclosure of the settlement amount.
Wait until the publishers start seeing insurance premiums skyrocket because of increased risk of lawsuits and having to defend the massive dam break of lawsuits coming their way once they see that people can sue and win. Pretty soon, so called "nuclear settlements" will occur, insurers will drop out entirely and the papers will be left with 1, 2, maybe 3 insurers that charge astronomical rates for stacked levels of coverage. Oh, yeah. Its coming. Regardless of whether this settlement was actually big dollars or not, the costs are going to hit the media and they'll have to change their model or be overwhelmed by the horde. Good, I says.
Based on previous reading (I will try to find the source) - there are numerous settlements between media companies in the U.S. and people who have sued them each year. Most we never hear about since they involve business people or non-entertainment people. The media companies and their insurers find it easier to settle than pay legal costs for a protracted period of time. I expect most settlements are small - more like low 5 figure than 7 figure. Even with the Sandmann settlement I don't see major media companies changing their ways - they simply view these settlements as a cost of doing business. I don't think their premiums will skyrocket to a level that forces them to thing about changing anything.