When you stop to think about it, it really is impressive that you represent an unbroken line of descent for millions of years of life.

Maybe. It feels a bit less impressive to me, since I see it as "I am one of billions of constantly dividing branch tips on a branch that didn't get pruned out of countless ones that did, all on a single tree from a single seed". The tree itself is impressive, the fact that every bud has a continuous path back to the roots isn't.