At many of the universities in question--large private universities, flagship state universities, etc.--EVERY professor has a Ph.D. (or terminal degree in their field).
But that is not really the important difference. When it comes to tenure, promotion, or being marketable for a job in the first place, often the most important thing for professors at top-ranked private universities is or state flagship universities is their own research and then their ability to successfully mentor graduate students and get them academic jobs (assuming an academic-oriented Ph.D. program). Undergraduate teaching comes way lower on the list, which is not to say it is unimportant or not cared about.
At a Community College, on the other hand, professors are typically not required to do ANY research of their own in order to get tenure or promotion. They also do not have to supervise dissertations or undergraduate theses. There is less committee work because there are no grad students and there is not a full on major in their field as in a bachelors institution. So they are able to teach more classes and do more grading because their position does not include many of the duties a position at a top research university includes.
To the original question, am a recent Ph.D. and TA-ed while a grad student for undergraduate courses as well as taught my own courses. As a TA I did all grading and held recitation sections (of ~25-30 students) for lectures that had about 500 students. In my department, professors who taught the lectures also taught one recitation course and did the grading only for that course. All of the TAs in my department were Ph.D. grad students in the department or in a closely related department. In the first few years, one would TA in the manner above. Then after one had the equivalent of an MA (3-4 years of grad school), one could teach courses completely on one's own.
Basically TA-ing is how a future professor learns to teach. It is expected when a freshly minted Ph.D. applies to tenure-track jobs that they have had a good amount of teaching experience already. Expertise is not generally an issue, because (at least in my experience) one is teaching say a 100 or 200 level class aimed at freshman and sophomores when one is a 2nd or 3rd year grad student (about MA level). One surely has the expertise for such a low level class. The real issue is learning how to be a good teacher and that is something you have to learn on the job. If future profs didn't learn this as grad students, they'd have to learn it in the first few years of being a professor--either way, some group of students will have to experience their first semester or two of teaching while they are learning.
I was never taught by a TA because I went to a very small liberal arts college where there were no grad students (though I think undergrads sometimes TA-ed in science classes, which I never took). Personally I preferred my experience as a student because of the smaller courses (~25 was a very big class for me whereas it is quite a small class where I did grad school), not because they were taught by professors instead of TAs.