If God has written our Destiny, then why do we need to live it?

There is some serious over-thinking going on here. Trying to understand Waheguru through our thoughts is not possible. Our processor can’t handle it. It will blow a circuit but it will never be able to understand how infinite and perfect Waheguru is. We aren’t perfect and so we can’t understand things like perfection and infinity.

Gurbani says:
ਕਰਤੇਕੀਮਿਤਿਕਰਤਾਜਾਣੈਕੈਜਾਣੈਗੁਰੁਸੂਰਾ॥੩॥”
About the extent or play of the creator Waheguru, only the creator Waheguru knows, or the valiant Guru knows.” (SGGS – Ang 929)
All we know is that Guru Ji has given us certain hukams (orders) and bachans (advice). All we can do is follow it.
Gurmat says that we have a choice in life. We are put in our situations in life (rich, poor, living in UK, born in India, or born in Canada, and in the particular families) based on our past karma. But our future karma is determined by our actions while we live in these situations.

If we had no choice, Guru Sahib would not have told us to do anything. There would be no hukams from Guru Sahib.
It is true that this is absolutely mind boggling to us humans how we have a choice but regardless God knows it. It defies our logic and our realm of possibility. But God isn’t bound by our limitations.
So the summation of advice: Stop thinking and start doing. All you and anyone else can do is follow Guru Sahib’s bachans (advise) and rehit (prescribed discipline) and the rest is up to him. Thinking about this will never clarify the issue though bhagti (loving devotional worship) might.