I was hoping some sort of recollection would come forth Mike, but my impression FWIW is twofold: one, that the original designs simply duplicated the concept of the many wooden 13 tonners (which also generally had bottom doors), and the second being of somebody having the vague idea that they could be used on coal drops in the absence of hopper wagons (albeit still with shovelling involvment, inside the wagon).
It's probably not coincidence that the LNER 16 tonners were the only pre-BR ones to not have bottom doors, and possibly the adoption of the LNER 21T hopper as a BR type was a significant factor in the decision to stop building 16 tonners with bottom doors (though that's only a theory, I'd have to check specific order/build dates to even begin to back it up). As for them being run together, again it's possible but in the typical 'train shot', it's difficult to tell; at the time that this would have been happening (if it was), they'd have been mixed in with ex-POs