javertim, if your "ultimate delivery medium" is a CD, then one thing is for certain: If you make your live recording at a sampling rate higher than 44.1 kHz, you will eventually need to convert it down. The processes available for this conversion range somewhat in quality, and can never improve the sound quality of recordings except by some kind of wacky accident; at best they can be qualitatively neutral while adding a little bit of noise.
Valid generalizations about which approach is best are therefore hard to make. However, I've noticed that this doesn't stop people from making broad statements; it doesn't even seem to slow them down. I view such advice with skepticism. It may be valid for the particular equipment or software that a given person is using, or it may just be what they'd like to believe they're hearing (or can hear), or it may be some of each. But if someone says "It sounds better if you start at 96 kHz" they can't possibly be speaking for all 96 kHz equipment, or all ways of converting 96 kHz recordings down to 44.1; that's simply not possible. And yet there is so much of that kind of careless talk. People only discredit themselves by stating that type of opinion.
There's one definite practical and (potentially) audible advantage to 24-bit live recording even when your eventual "delivery format" will be 16-bit. The advantage is that, with suitable equipment used correctly, you can set your recording levels conservatively. You can avoid accidents if an unexpectedly loud sound occurs, without giving up the full dynamic range that 16-bit recording offers.
What I mean is, if I'm recording live at 16/44.1 and I'm going to deliver a CD to the client, then I really will be trying to get the peak recorded levels to land somewhere between -1 dBFS and maybe -3 or -4 dBFS. If they're lower, people can't help imagining that they might have guessed more accurately themselves, while if they're higher, then there's distortion (overload). And that makes for a somewhat tense situation because, what if I'm wrong? I'll have to change the levels during the recording, keep careful notes, and compensate afterwards during the transfer to the final product. That's a fair amount of extra hassle and the dynamic range of the product won't be ideal.
With 24-bit live recording and 16-bit delivery, though, I can allow (say) 7 or 8 dB headroom. If the performers want to surprise me, they can go right ahead. In the end, either way I'll reset the gain and dither down to 16-bit, and come out smelling like a rose every time. I like that.
--best regards