Why is it thought to be "more appropriate" to deny the existence of a mental illness? Your friend would not be in hospital if he were not ill. Therefore, it is appropriate to wish that he gets well.
I think refusing to acknowledge the nature or even existence of his illness is feeding into the stigma surrounding mental illness.
If he had cancer/brain tumour/appendicitis or any other physical illness you would wish him to get well.
Why would you not wish him to get well from a mental illness?
Your card so your choice. Personally would send a blank "Thinking of you" card and put something like "Know you're going through a rough time at the moment. Thinking of you and hope to see you soon."
A lot of psychological conditions are ones that you don't 'get well' from - rather you learn to manage them and they stay with you for life - I have one myself.
Same here, Andy. Depression and anxiety for over 30 years. You do 'get better' by learning how to cope with this ailment, but i don't think i'll ever be rid of it.