That there were bills in Congress at the time to make football illegal, certainly makes TR's efforts part of a larger effort.  Here's a headline from the NYT at that time:

FOOTBALL IN 1909 CAUSED 26 DEATHS;

Highest Total in Many Years and Almost Double That of 1908 and 1907.

70 MORE SERIOUSLY HURT

Chicago Tribune's Figures Show That Majority Killed Were College Players Trained by Expert Coaches.

This was a national political issue at the time that was far more than TR's meeting with college football officials, though that was crucial in saving the sport.  

Without the pressure in Congress and TR's effort there would have been no NCAA.  Therefore, to my mind, that makes the NCAA a child of the Federal government.  They were responsible for its formation. TR actually had said publicly that he was going to recommend the abolishment of football, but then changed his mind and called a meeting of college football leaders in the White House.  

Take a look at this article:

http://www.history.com/news/how-teddy-roosevelt-saved-football

Here are the two ending paragraphs:

Following the [1905] season, Stanford and California switched to rugby while Columbia, Northwestern and Duke dropped football. Harvard president Charles Eliot, who considered football “more brutalizing than prizefighting, cockfighting or bullfighting,” warned that Harvard could be next, a move that would be a crushing blow to the college game and the Harvard alum in the Oval Office. Roosevelt wrote in a letter to a friend that he would not let Eliot “emasculate football,” and that he hoped to “minimize the danger” without football having to be played “on too ladylike a basis.” Roosevelt again used his bully pulpit. He urged the Harvard coach and other leading football authorities to push for radical rule changes, and he invited other school leaders to the White House in the offseason.

An intercollegiate conference, which would become the forerunner of the NCAA, approved radical rule changes for the 1906 season. They legalized the forward pass, abolished the dangerous mass formations, created a neutral zone between offense and defense and doubled the first-down distance to 10 yards, to be gained in three downs. The rule changes didn’t eliminate football’s dangers, but fatalities declined—to 11 per year in both 1906 and 1907—while injuries fell sharply. A spike in fatalities in 1909 led to another round of reforms that further eased restrictions on the forward pass and formed the foundation of the modern sport.

So, it seems clear that without the pressure from Congress and the intervention of TR there would have been no national collegiate athletic association formed at that time.  Therefore, it's not unreasonable to say that the NCAA was a child of the Federal government.  I'm not playing semantics.  It's a reasonable way to summarize the actual facts of what happened.  You are welcome to describe another way, if you prefer, but my statement is not an inaccurate way to describe the creation of the forerunner of the NCAA, without which there wouldn't have been an NCAA.