In reply to Misha:
I'm trying to think of routes where this might be better but honestly can't easily think of any, especially as you would need to climb with your skis (or haul them somehow?).
If you have to do that, then surely the level would be such that you would be fine climbing in your touring boots? Especially something like the Dynafit TLT series or the new Scarpa stuff.
To be fair, ten years ago (or maybe even 5) this question was much more pertinent, as it was a much bigger differential between climbing/ski touring boots as now ski boots are so much better for climbing in, especially as ski mo races have taken off and the technology tricles down. So arguably, there really is no need (in the alps, rather than alaska) to ski at all in climbing boots.
For instance, the lads who did the walker recently (and some polish dudes who went to try the colton mac the same day) left their skis at the bottom to pick up later (or get someone to skin up with you and take them down), and that's exactly the type of situation you speak of.