Tbh I don't really see what the issue is here. So he is scared of flying? Thousands of people are. I know I am. I tried it once, had a bad experience and if I go near an airport break out into a cold sweat.
I can ride a bike, don't don't really enjoy it, so don't do it.
I am terrified of the tube and trains. I get horribly sea sick so don't bother.
Due to nearly drowning at 5 I will not go near a pool.
My dh accepts me for who I am, phobias and all. Yes he had a hard time getting used to the fact that there are no foreign holidays and we had to holiday in the uk (shock horror) but the way I look at it is the uk tourist industry needs help, and that is my excuse 
Fortunately my dd is far more confident than I ever was and she gets that from her daddy. I try not to pass on my insecurities, but I am of the belief that if you are naturally anxious that is the way it is.
You can accept him for who he is or not. Don't try to change him (as an ex of mine tried to) as that will lead to unhappiness on both. He will change if HE wants to. As my therapist told me, only you can make that change, but you have to want to deep down. Some times it is just easier not to face things. His phobias that you list are a little limiting, but hardly life threatening?