It’s not an area I have an interest in. I just came across the comment and posted it.
The best I can do is suggest two things. Firstly, notwithstanding the heading, Army Surgeons, viewtopic.php?f=27&t=10444
, contains references to a number of articles on various 'medical' subjects over a seventy year period. You can search the text of a number of them. I’ve spot checked a few. Wrt “what training they had for the role”, it seems not much more (as in: if anything) between the comment that aroused your interest and 1902. See Report On the Medical Arrangements In the South African War
, p 76.
Wrt to ”when the bandsmen's role as stretcher bearer/first aider first came about”: As evidenced above, if you explore the articles <1868 you might get some idea.
Secondly, if you’re interested in the role of bandsmen & / or stretcher bearers in the 1868 Abyssinian campaign you might look at discussions of the Campaign per se in 1868 Abyssinia, viewtopic.php?f=18&t=7158
, this Forum more generally, and Google et al.