Also, I feel it would have weakened the idea of Sauron being this omnipotent being of pure malice that he would resort to purely physical means of defeating his enemies. If Saruman had traveled all the way to Caradhras and tried to chop Frodo into little pieces, instead of causing an avalanche, wouldn't that have been somewhat cheap?
Well, despite the quirky and rather staid sensibility of its author, the LotR is a masterful narrative. The way Sauron is handled is one example of that: always in the background, always looming, never physically present, with even his servants and emissaries appearing as terrifying and overwhelmingly powerful.
So, yeah, having Sauron appear in physical combat would certainly have lessened his stature, but it is the kind of temptation that a director of Jackson's quite literal sensibilities would be likely to succomb to. (He does in fact succomb to it, at the beginning of Fellowship, but that is less of an issue because it is a flashback to a previous era. I've always found Sauron's appearance in these scenes to be rather ridiculous, though, in any event.)
For this reason, I think it was certainly the right choice to remove Sauron's battle with Aragorn, though Sauron is far more effective in the books than in the movies anyway, for a variety of reasons.
One is that we learn more about him in the books (and more about his servants such as the Nazgul and the Witch King). Another is the red-eye radar in RotK which is an effort to capture the power of Sauron's gaze on a weakening Frodo, but which comes across as rather silly. Blatantly so, actually, I remember laughter breaking out in the theater when I originally saw this, which was certainly not the desired effect. Again, too literal. That's the weakness of Jackson's adaptations overall, I would say, though that same quality doubtless contributed to their success.