Ah, well, let me say that this is an interpretational issue, which means it is really up to anyone. For me though, you have to trill the whole time. Explanation:
Let us do some analysis. The Waldstein sonata represents the "new way" of Beethoven piano composition, together with the Appassionata it forms the inner core of his middle period. Obviously, he has changed some things in his writing. Look at the first movement. The first theme is a steadily repeated C-Major chord, no melody, just a sound sphere, a "Klangfläche" (sorry, translation, anyone?). Also notice the "drum roll" at measure 142 and so on. What I mean, Beethoven uses the piano as a mean to produce sound, not to produce a certain melody (he does this too, of course, but these other techniques become quite predominant, considering they have not been used before). Even the runs (bar 9/23) are more a mean to display the whole "attitude" of this movement, and that is break-up, a start, a real beginning (also first theme!). The entire movement, even the rather moderate second theme, represents this impatience, again and again you find a composed accelerando in the accompanying figures, and of course the grande scene at the end of the development (after the drums...), you know, the fanfars. Anyway, this movement, for me, aims at only one thing, and that is the entrance of the Rondo-melody. Incessant on-going of the music in the first movement, the only destiny: the future. This changes witht the second mov.
Here, the impatience changes to quit and pleasant anticipation. That is also the reason, why the Introduzione has substituted the Andante Favori. Not because it was too large, that would actually fit better with the proportions of the other movements. No, it is just that after this first movement there is no room for other ideas. The music urges to this melody, now, in the second movement, it realizes that the moment is not far away and therefore it can only wait.
Now, with the beginning of the Rondo the moment is there, and we hear this incredible melody. And from that point on, Beethoven tries to not let it go, forever. The whole movement seems to me the "taking hold of the moment", an eternal "not letting go". See how many times the theme is reintroduced, nothing changes with it, there is really not happening much in the whole movement (besides the second parenthesis (is this the right word?, you know what I mean, measures 175-312)). So, what to do with the melody? And now we are back to the original question. I agree, it sounds better to leave off the trill, considering you WANT to bring the melody out. Let's see what Beethoven does with it. The first two times he brings up the theme, he uses a strange RH accompaniment. Here, the melody indeed "substitutes" the bass figures. But then, where normally something would happen to the melody, Beethoven just transposes to the F. Still, he uses the same patterns in the RH. Then, the melody is heard in octaves in the RH, whereas the LH has another accompaniment pattern, much more interesting, scales, more movement, etc. Most important, the accompaniment is heard the whole time, no melody note substituting anything. Anyway, over this, the melody still reigns, but it is already the 8th time you get to hear it in bar 43. After this, there has to happen something "newer" than just another pattern. And indeed, Beethoven (re-)introduces this really cool idea of trilling a note under the melody. Mozart or Haydn didn't do this, it is part of his "new way". And, in the LH he comes up with an even more powerful accompaniment, with 32th notes up-rising from the depths of the bass. So, what do you think does Beethoven want? Does he want you to make the melody stand out another time (9th and 10th) or do you think he wants you to show his new super-cool trill idea? This trill is essential to the movement, it electirifies it, lets it vibrate and begin to "glow". Again, as many pointed out, it really does not make much of a difference. And you can show the trill even with substituting it with the melody notes. However, I think this dissonance created by the trill brings this movement its sublimity.
In the further development of the movement you see more proof for Beethoven's unwillingness to let this melody go. In the second parenthesis there is this huge relaxed end part, kind of improvised, then the theme again, the first parenthesis repeated, but with another LH pattern, and then, no, not the end, but a grand Prestissimo, repeating the melody countless times again. It is here where I find my ultimate confidence on the matter of how to trill. At the end of the autograph Beethoven shows you how to execute the Prestissimo trill, namely either playing triplet quarter notes (six per bar) or triplet eight notes (12), with the trill beginning on the upper note (illustrated as well with the fingering at measure 485: 21). And that does mean, trilling ALL THE TIME, no leaving off of the trill (it also contradicts the idea only the G should be played while playing the melody note for the sake of consonance, rather, it shows that you really have to create this dissonance), in my Henle editions you have the Ossia-picture where you can see the trill-triplets under the melody note.
You could now argue that the Prestissimo trill is another story, because after this whole movement you really do not have to bring out the melody as strongly as previously. Well, I disagree, but that is just my opinion.
I realise that this "interpretation" attempt is nothing near complete. It just shows the broad view I have on this sonata, and lacks many missing links (I could provide, though). So please don't rant that I have missed this or that. If you disagree with a point of mine, then we should have a discussion.
Topic: Help me with Waldstein