Helps to consider the recording, processing and delivery steps as completely separate aspects with different requirements, each following the other. In other words, it's not about processing.
Recording- All else equal, if the live signal dynamics fit within 16 bits, the recording will not suffer from being made at 16 bits, nor will it be improved by recording at 24 bits. However, it is often easier for the recordist to make the recording at 24 bits, since doing so is likely to increase the allowable dynamic range to somewhere between 17 to 20 bits or so (depends on the performance of the specific recording gear used), making the setting of recording level easier with increased safety-margin. Personal conclusion- makes sense to default to recording 24 bits, retaining decent level setting for the reasons indicated above.
Processing- Best practice to do your processing in an editor which uses a sufficiently larger "calculation space" than the recording itself. 32 bit floating-point mathematics provides this for both 16 bit and 24 bit recordings. If the recording fits in 16 bits, providing the same data to the editor in a 24 bit file rather than a 16 bit file file won't make it work better or do its calculations more precisely. However, doing the calculations in a calculation space which provides increased precision over that of the original data can do so. Personal conclusion- makes sense to edit in 32 bit floating point regardless of input and output bit depth.
Delivery- All else equal, if it fits.. I think you get it. Note that this is what folks other than the recordist actually get their hands on, and as such, justifiably or otherwise, what they are likely to be most opinionated about.