[details snipped]

The point I was trying to make earlier (which came across as "nobody
should be coding their own Newton iterations")

is essentially this -
surely iterating a relation until the value you need converges is a
common operation, and libraries exist that do this for you?

FWIW, a quick google search came up with scipy.optimize.fixed_point,
which is, AFAICT, the equivalent of your iterate(). (You probably know
better than I do if that's the case).