I've found implementing test as I go along writing a new API help code quality and speed in two ways: I catch my own logic bugs early and the self-feedback I get from writing the tests helps me to write a more usable interface.

I never catch all the design errors with non-coding design work - this gives me the high level view which is critical - but there are always some logical errors and bugs I've failed to model correctly which early tests catch.

But yeah, the optimal design-implememt-test probably depends probably very much on the implememter and on the problem to be solved.

The third meta-advantage is that if I'm writing hobby code for myself and the profject is large compared to the weekly time allocations available in my sparetime I get something to help me remember where I was (say a month back) and gives me something to compile and run immediately (which is a huge motivational booster for me).