Regarding secular studies in high school, the only reasons it is allowed is either because education is mandated by State Law (in New York it is until age 17), or simply because if they did not have high school education in the Yeshivas, parents would simply send their kids to worse places to get it.
But it is definitely looked upon not as a l’chatchilah, but rather as something that is annoyingly necessary in the current environment.
Today, there are a number of High Schools in America – particularly in Lakewood – that do not teach English. Also, many Yeshivos do try to reduce the amount of secular studies as much as possible, through knocking out the last semester of English, and a number of kids are leaving HS early to enter Bais Medrash.
Rav Chaim Segal ZT’L, the Menahel of the High School at Yeshiva Chaim Berlin was once told by Rav Shach ZT’L that if possible, he should not be teaching English studies. In Eretz Yisroel, almost all Chareidi Yeshivos do not have English at that age. Rav Aharon Kotler ZT’L made some kind of commitment not to allow English studies on the HS level in Lakewood. The exact details, and if this was actually a Takanah or merely a preference, is not clear and depends who you ask. In any case, Rabbi Elya Svei, Rosh Yeshiva of Philadelphia and a student of Rav Aharon’s, was asked why he allows English in Philly if Rav Aharon was against it. What difference can there be between the town of Lakewood NJ and Philadelphia PA? Reb Elya answered that he has no choice, and that currently, the Baalei Batim would not send their kids to the Yeshiva except under these circumstances.
Is any of this the ideal? No. It is not. Is it justified? The schools say it is, as they have no choice. But the point is not what the Jews do, its what Judaism wants. Everyone agrees that it would be a higher level, a preferable situation if we would indeed not learn English even at the HS level, at least not beyond what is necessary to survive. Nobody claims it is an ideal.