First, there are vanishingly few academics in your (unspecified) part of the world that have done a PhD + postdocs in math, left academia, and then successfully pursued a career in physics research. So, I think you are unlikely to get any super helpful answers. We could speculate (and some of the existing answers already have), but as a PhD yourself, your ability to speculate is probably comparable to ours.
I assume this is why certain users voted to close. That said, I am not convinced this decision was correct. It's possible (albeit unlikely) that we have a member here who has been in physics at a high enough level for a long enough time that could give an authoritative answer. It's true that there are probably no statistics or references that anyone could provide, but "personal expertise in academia" is routinely used as the basis for answers here. Again, I think it's unlikely that you'll get an answer like this, but we do not close questions just because they are unlikely to find an answer.
Anyway, I'll leave this as it is for a while to see if we get any other points of view. To the other points you raise....
I feel like real examples of transitions would be actual data points and absolutely not opinions.
Perhaps not opinions, but we don't really take "poll" questions. Our format is that we vote for the best answer; if you ask for examples, then all answers are equally good. But you are correct that if you ask for an authoritative answer, one could provide an answer based on a single data point (it would then be for the voters to judge this type of answer).
Would contacting professor working on the topic I'm interested in help me make progress towards that aim?
Could doing student projects with a professor work as a preliminary step (and learning exercise) towards a real postdoc?
I doubt that without guidance I could make any meaningful contribution to the topic, but maybe I'm wrong to think like that?
Is there any concrete step I could do to achieve my goal of becoming a researcher in say, the mathematics of particle physics?
Note that you are asking four different, though related, questions here. I think it's okay in this case, but it is a bit of a "red flag"; you might consider making your question a bit crisper.
