https://www.ncbi.nlm.nih.gov/pubmed/17721406
Removal of adenoids reverses sleep apnea
There are more papers that show that it even reverses ADHD like symptoms.
This is causation.
I think there is a tendency on this board to jump to bold conclusions.
Regarding your quote: It says that there is a multitude of causes for hypertrophic tonsils, among them infections.
These enlarged tonsils or adenoids cause mouth breathing. As a result the child develops sleep apnea.
To sum it up:
Adenoids in children can cause sleep apnea. It is vital to diagnose and treat it early as it will have an impact on facial and dental growth. In more severe cases it will lead to failure to thrive.
Of course breathing through your mouth is a symptom of adenoids and not directly a cause of sleep apnea.
But it is an indicator that something is wrong with your airway and is directly involved with it.
EDIT: Alright, there seems to be some major confusion. Adenoid growth leads to nasal obstruction. Any kind of nasal obstruction increases the risk for sleep apnea, especially in children. Turbinate hypertrophy as another example:
https://www.ncbi.nlm.nih.gov/pubmed/23070869
Fact is: If your child is a mouth breather, there is a problem.
Perhaps you guys can specify what you mean by breahting problems.