Designing new protocols and constructions using primitives like AES and DH is exactly what cryptographers are talking about when they refer to "rolling ones own crypto".
Look at the description of their protocol, particularly the handshake, which negotiates number-theoretic parameters, and try to make a list of all the checks they'll need and the additional protocol that they'd need to specify just to make the handshake secure.
Not only that, but the crypto in Telegram is idiosyncratic; for instance, this is the only system I've ever seen that used IGE mode.
Look at the description of their protocol, particularly the handshake, which negotiates number-theoretic parameters, and try to make a list of all the checks they'll need and the additional protocol that they'd need to specify just to make the handshake secure.
Not only that, but the crypto in Telegram is idiosyncratic; for instance, this is the only system I've ever seen that used IGE mode.