I doubt you'll see any significant long term reduction in homelessness or poverty with such a program because there's little to no evidence that indicates that biology is a major cause. Some of the strongest predictors of homeless rates in US cities are availability of low income housing and of mental health care [1].

There is some evidence that homelessness & poverty increases the likelihood that the next generation is homeless/poor but that would be a secondary factor. The moral & political cost of a forced sterilization program is indescribably immense in the West and would need to promise similarly immense outcomes which are nowhere to be seen.

[1] https://www.jstor.org/stable/800641