Physicists consider classical mechanics to passe' today. It does not have cache'.
It was not so in the 19th century, but today they consider there to be nothing fundamentally new left to learn in mechanics, so they have moved on to new things that include buzz prefixes like "quantum" or "nuclear" or "sub-atomic" or some such. And they are correct in that the general principles of macroscopic mechanics are well established; there remain, however, some absolutely great problems to be solved.
EM and TAM are essentially the same thing. Different schools use different terminology, but the difference is nothing that one can ever define. At the University of Texas at Austin, it is the Dept of EM & Aerospace. At the University of Illinois (Champagne/Urbana) it is the Dept of TA&M. The UT-Austin department is a bit more space oriented, the UofI department is a bit more structures oriented (or at least these statements used to be true; I'm out of date now.)
As to why relatively few colleges offer majors in Engr Physics or EM, it is mainly that it is a niche major, low demand. There is not a great demand for this major anywhere, and most schools simply cannot afford to offer a major in a low demand major. UT-Austin offers an Engr Science degree (i think they still do) that is comparable to an Engr Physics degree.
If you want to deal with real world problems, either in things like civil structures, or machinery, you can get the necessary education in either an ME or EM department. For something like this, I would not even consider a physics degree. Engineers are the ones who deal with the real, macroscopic world that we live in every day.