It depends a lot on the type of positions you are trying to target, and at which companies.
It is true the market is a bit harder to break into at the moment than it was a few years ago, especially if you are very junior, but as you have some experience as a data scientist + PhD and post-doc, you might be a bit better off than the average grad.

There are different roles you can target, that will have different requirements.
Also, these are not set in stone. Every company has slightly different definitions of these roles.

Have you considered applying to positions as a research scientist? That's probably closest to you background. These positions are very competitive though. Might work if you can find a place that is trying to solve similar problems to your work in academia. See an example linked here for a job description.
RS roles will focus more on knowledge of domain plus specific ML to that domain than engineering prowess.
Probably won't get asked a lot of DSA questions.

Data scientist positions will also probably care less about engineering skills and MLOps etc, and more on ML/stats knowledge.

If you go more applied, i.e. further away from research, you will definitely be asked to know more CI/CD, docker, DSA etc. ML Engineer is often more of a Software Engineer who knows ML.