Go look at Japan to see what happens when you reduce both births and immigration.
An aging population, combined with falling fertility rates and no immigration is a disaster waiting to happen.
You need fresh blood to keep an economy going - they form the backbone of the workforce (read: taxes) - and in general, they're not scared of trying new things.
No, that's what's going to happen in Europe.
Japan will just have to make some deep, technical changes to budget and taxes in a culturally and ethnically homogeneous civilization, not have to deal with social upheaval.
Sure, so the inheritors of my civilization might share like, only 99.95% of my DNA instead of 99.99%, but so what?
Yeah, it only gets replaced.
> More diversity, more innovation.
Sure, sure.