indeed the regulations didnīt told the officers to do some deadly "sports" in front of their units as well.
>...it seems to me highly unlikely that a company level officer would be mounted on the battlefield.<
Itīs unlikely, but as far as I remember, the tirailleur commanders of the Prussians should have been mounted in the first time in 1813, but all bataillon officers were ordered to fight fight on foot, when their bataillon started skirmishing, after a short time.
I think it was most probable possible that maybe only the british flank company commanders, which were designed for beeing detached, sometimes had their horses with them. It could have been a matter of exaggerated sports attidude as well. For the mentioned brevet major, itīs the question, why he had his horse with him, and just gave it to a drummer boy (not leaving it behind the baggage! Did he had the command over a half bataillon? If yes, it was insubordination to leave his post. For some officers on horseback other things played a role. They werenīt able to serve on foot anymore, but refused to leave their posts also. The result were mounted company officers. The same thing is told about Cpt. (Maj.) Menzie, about whom the text says he was lame due to his wound received in Spain, so he might have been able to fight standing, but not marching. There were often partially handicapped officers in the ranks.