Currently in the United States ground beef can be sold under the label "100% ground beef" as per the the USDA Food Standards and Labeling Policy Book (August 2005); however, there have been a number of recent claims that lean finely textured beef (LFTB) or "pink slime", not to be confused with mechanically separated meat which is not considered safe to eat due to Bovine Spongiform Encephalopathy concerns, may not be safe to eat and proper studies have not been conducted. Is there actually any validity to this claim or has the issue been over-hyped by the media?
Let's break down the core issues raised in the cited New York Times article (which remains my main reference for this answer).
What is important to this question is that none of these points make any claims about the long-term danger of the meat. The only real health concern raised is in the short-term, viz. is the meat contaminated with Salmonella or E. coli, both of which can cause severe food-poisoning, especially in children?
And it is clear from the article that the answer (from both officials, schools and the manufacturer) is yes, bacterial contaminations do, very occasionally, occur in the raw product. Beef Products Inc deny that it is more common than from other suppliers.
Now, whether you consider the risk of infection too high for your children is a personal choice. (Remembering it happens in other meat too, and can be ameliorated by proper food preparation, including cooking the food adequately.)
I haven't tried to challenge the reporting from the article, because I believe the article does not directly make the claims that the title suggests, so there are no substantial claims that have been called into doubt by the question.