At least for me personally, age means nothing, only ability.
and age has a direct connection to ability in many cases.
let's take Business English:
Is a 23 year old guy, fresh out of college who has never had any sort of professional career and probably never even a full time job able to relate to a 33 year old guy with an upper management position? Does the 23 year old have his own business experiences, strategies, intuitions....?
he talks about his boss riding him every day, you've never had one.
he wants to discuss his annual performance evaluation, you've never had one.
he wants to talk about corporate culture, you have never been exposed to it.
he wants to discuss business strategies and how they work in practice, you've never practiced anything.
You can talk words, but you can't talk business and it's a legitimate request when a student asks for someone "older with more experience".
Think of it this way: is it necessary for a language school director to have language teaching experience? no. the job doesn't require you to teach, rather your main responsibilities are to sell contracts, keep the peace, run the school in an orderly way, manage your teachers and other staff and be a good spokesperson for the school. BUT.....could and would having previous language teaching experience make you a better director at a language school? most certainly.