Christensen is a much better defender than Boateng was at his age. Boateng was a decent prospect but nowhere near as good as Andreas was at his age. He improved significantly under Pep at Bayern. I wouldn't really call him a new school centre back because he has good ball skills. He has a very classy and elegant way of defending, that some old school defenders like Beckenbauer or Maldini for example had too. He wasn't voted BMGs POTY just because he has nice skills on the ball, he's actually a smart defender too. Conte will do wonders with him.
You talk about Cahill being better in the air but that stat proves that it wouldn't be a downgrade per se. The guy will get bigger. He has the right frame already, and will naturally put on more weight in the coming years. Also, Cahill can be quite stupid when it comes to marking in set pieces, as we've seen on numerous occasions in his career.