I suspect that it would depend on one’s definition of “Christian”.
Members of the Eastern Orthodox Church consider themselves to be Christian.
Roman Catholics consider themselves to be Christian.
Methodists consider themselves to be Christian.
Baptists consider themselves to be Christian.
So do Christian Scientists and Jehovah’s Witnesses.
If you have a particularly narrow definition of “Christian”, you might not fit in with those of Christian faiths that you do not recognize as “Christians”. While each Christian denomination thinks that they’re the best, the one true religion, it’s kind of narrow minded not to recognize that denominations other than your own call themselves “Christian”. At the type of schools you mention, it’s preferred that students respect each other.
If, instead, you are using the term “Christian” because you are uncomfortable making the discussion about a specific denomination, it might be less of a problem fitting in.
I think you should widen your search a lot, Focus on schools that support your values and/or have active student groups for those of your faith tradition. There are many wonderful boarding schools that are academically challenging. Don’t just focus on Harkness - many schools have something similar to Harkness, sans trademarked table.