I'm not sure I want a mount for home and a mount on the island, though. I mean, when you get the "horse flute", then what? Does it summon the ostrich when you're on the island and the horse when you're at home?
For that reason, I think it'd be better to stick with one mount... and possibly have it be an unlockable upgrade to the stable? It could then allow you to interact with the building to select between mounts in that building. Perhaps the upgraded stable would also allow you to send your horse home instead of it only going home without you at the end of day? Perhaps the ostrich mount could be quicker, but skids to a halt too instead of instantly stopping like the horse does?
I mean, I get why folks would like having a mount on the island. The horse flute is literally the first thing I buy with my Qi gems... but with the horse flute currently being hidden behind you getting a certain number of golden walnuts and then doing a couple Qi quests to get the gems to afford it, making a mount available before you do those things would pretty significantly change the dynamics of how quickly you can get things accomplished in the early days/weeks of you being on the island. While I like the idea of riding around on an ostrich, I'm not sure I would like it being implemented as an easily-attainable mount on the island and/or only for the island.