I'm afraid that Audition won't do this - even though on one level, ryclark notwithstanding, it's strictly logical as an operation goes, I suppose; as it is, it appears to delete a section of the other channel even though it's deselected, and that shouldn't happen. Hmm...
But let me try and explain why: There's a major difference between Goldwave and Audition, and that is that as you've noticed, Audition has a multitrack view. And because of that, any sort of operation that you'd think you might want to carry out like that has effectively been shunted off to that view. It makes sense, too - if we take your example, what you'd do is go to Edit>Extract Channels to Mono Files, dump them straight into a multitrack session, and then you could slide one against the other easily, whilst monitoring the effect, making it much easier to line the channels up where you want them. Make a time selection, export it and you're done.
Now you may say 'but I want to do this in Waveform view'. And that's fine - put in a feature request. But I can tell you now that the only logical reason for implementing it is that strictly speaking, Audition isn't doing the thing it's indicating that it will - it's supposed to be leaving the deselected channel alone and it isn't. But in principle, doing this sort of operation in that view isn't the way Audition is intended to be used, simply because it is, at heart, a multitrack editor.
There is another snag too. You have to bear in mind also that you're only looking at this in stereo, and Audition supports multi-channel files, not just stereo ones, which Goldwave is restricted to. Whilst I'm not saying that it's impossible to implement this, I suspect that it might not be quite as simple as you'd possibly expect...
(amended slightly...)
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