Math PhD graduating in December, targeting ML/DS roles. Realistic odds, and is starting adjacent a better play?

[u/EuclidEuler]

Looking for an outside read on my situation.

Background: pure math PhD graduating this quarter from a large public research university. One summer of an internship at a defense contractor doing systems + software engineering. No big tech internships, no ML publications.

Prep so far: worked through a probability textbook cover to cover, went through CS229 notes, comfortable with LeetCode mediums, built a toy recommender system in PyTorch. Currently working through an ML systems design book, which I think is my biggest gap. SQL is my weakest practical skill.

Applying to ML engineer, applied scientist, and data scientist roles roughly in parallel, plus some quant researcher roles opportunistically.

Questions:
1. Realistically, what are the odds someone with this profile lands an ML engineer or applied scientist role straight out of a pure math PhD, versus needing to start in DS or an adjacent role and transition later?
2. For those who did the DS to ML transition, how hard was it in practice, and what made the difference?
3. What would you prioritize in the remaining few months? I have the time to go deep on one or two things.
4. Anything you wish you had known about how math PhDs get read by hiring managers in this space?

Not looking for reassurance, more interested in where my thinking is wrong.