This might be true. I'm not
old, so I'm not certain.
Birthdays used to be wonderful days, where you celebrate you adding another 365.25 days into your life experience. Especially when you are below 10. Each year is a significant portion of your life. That's why birthdays as a kid were so important.
As you get older, you start to care less and less about this magical day that used to be so important. Probably because with each passing year the amount of time between each birthday lessens and becomes less significant, in your mind, time passes faster.
I don't know what mid-age crisis is. Probably world-weariness. I would guess at that age, people who suffer from that crisis would dread each birthday instead of welcoming it, for it signifies another year lost that could be used for "better" things.
My grandfather likes birthdays. He is nearly 80. He has enjoyed life, so he is prepared to pass away. Another year to him is nothing but more time spent enjoying existence. That is probably because he had a fulfilling life, despite all its difficulties (communism in China, scratching out a new living in Hong Kong, and so on). That would be what I want in old-age. Enjoying it while you can.