as a public course player, I've always imagined that private courses were a paradise of timely play, and that rounds there are almost always completed in 4 hours at the most....because the skill level across the board was generally higher, because there were measures/policies in place that all the golfers knew about to keep the play moving, and because members had to see eachother often and so there was 'social pressure' that mitigated slow play.
I played for ten years entirely at public/semi-private courses and have since been a member at two private ones. Overall I'd say there is a slight pace-of-play improvement at the privates but course-to-course variation in how the course is laid out has a much greater effect on pace. My first private club had a 90's-vintage housing development course that was hilly, slow to walk and not particularly fast to play with a cart. Lots of cart-path-only days due to clay soil. Weekend pace of play was very similar to the public course I learned to play on because the public one (which did 4-5 times as many rounds easily) was built in the middle of nowhere in the 70's and very compactly laid out.
My current club has a 60's Ellis Maples layout that is ideal for walking and with a pretty good flow to the entire facility (BTW, I believe the amount of wandering around the parking lot, clubhouse, etc. before and after the round sets the expectation for pace of play in a major way) and our pace of play is similar to your expetations. Other than the very busiest times of peak days you can generally finish in four hours. During the winter it's 3:15-3:30 for threesomes and maybe 3:45 for foursomes. This time of year the course is generally packed and anywhere from 3:45 to maybe 4:10 is typical.
My club has no "measures/policies" relating to pace of play. Simply informal peer pressure to not hold up the other folks who you're going to be sharing the club with for the next few years. As at all courses in my experience, any sort of organized competition or event brings with it a glacial slowdown in the pace of play. I don't know if peer pressure is somehow suspended on "tournament" days or if it's the universal cart-use requirements that typically accompany those occasions but there's no such thing as a sub-5:00 round during a "Couples Tournament" or "Labor Day Scramble", not to mention the Club Championship where everyone turns into Jack Nicklaus.
Fortunately, we have 27 holes and none of the three nines is considered undesirable. So there's usually a way to avoid the backup if one chooses not to participate in this month's hit and giggle. Finally, there are probably fewer raw beginners at a private club than a public one so in that sense the quality of play is better at the low end. But I don't think the possible larger number of plus-handicap sticks at a good-player's club results in any net gain in pace of play. In my experience the +2 guys aren't necessarily faster than the 6-handicappers so the only difference that factor makes is if your Sunday afternoon round avoids the hordes of "first time ever" golfers that descend on some public courses.