I'll try to answer your question through addressing each sub-question:
would, say, a modern philosopher with a PhD be well-versed in modern
A philosophy PhD, much like any other higher degree, doesn't always describe the specifics of the underlying focus of study. Not all philosophy PhDs will be well versed in modern psychology, but those focused on subjects like philosophy of mind, consciousness, cognitive science, et al. certainly will have addressed overlapping subject matter during their education - be it history, theory, etc.
There are concrete examples of modern philosophers who are well versed in psychology, both from a theoretical and applied standpoint. Two examples that come to mind are Noam Chomsky and Daniel Dennett; both of whom have (and continue to have) lasting impacts in the field of cognitive science and psychology.
There is a distinction, however, between being "well-versed" in the field of psychology and being well versed in practicing psychology. So, if you're question boils down to "are all philosophy PhDs qualified to practice psychology" the answer is, "no." A physics PhD might be well versed in rocket propulsion, but it certainly doesn't qualify them to repair a space shuttle.
Since philosophy inevitably investigates the mind and the behavior of
people, and that most philosophy degrees require courses that bear
tremendous resemblance to those that a psychologist takes, do the two
have noticeable overlap, or is philosophy more primitive and
theoretical in this regard?
Psychology wasn't a separate area of study until the late 19th century; meaning, prior to that date, the study of psychology was considered a branch of philosophy. You might be surprised to learn that natural science was also considered a branch of philosophy until they began to split in the 16th century.
In order to understand modern psychology (or philosophy for that matter,) its important to learn the historical context that's brought the field to it's current state. So, there may be some overlap in who you read in introductory philosophy and introductory psychology courses.
That said, the answer to this question is emphatically, no -philosophy and psychology degrees DO NOT have a noticable overlap. The reasons why you read Descarte for a philosophy degree is dramatically different for why you might read him for a psychology degree. Same could be said for Spinoza, Kant, Hume, et al.
Philosophy is not a "primitive" or "theoretical" version of psychology, it's more like the study of "challenging assumptions through logic and critical analysis." It can be applied (and is) to math, science, art and a litany of other fields of study - including psychology. Psychology is the study of the mind and human behavior. They are very different fields of study.