In a General Q&A livestream (July 2019) the real reason why it is not possible to introduce mounts into runescape is revealed:
"Will Jagex ever release mounts?
R: No. There are so many issues with mounts. If you can help solve the two biggest problems with mounts then we will. (i) The game has been designed for 1x1 characters which includes doors and agility courses; (ii) Run speed is the fastest the player character can go. What would a mount do, go at run speed? Or teleport a few squares. That's pretty much the only two options you've got. A lot of people also want to fly which has its own problems."
My opinion: In fact it really seems difficult for mounts to be launched even in the long time. But it's still one of the ideas I'm not really willing to give up. So I want to keep it there, maybe improve it and even keep you up to date on whatever position Jagex might take on it.
For now contain yourself with this list of iconic runescape creatures that I have selected for possible mounts:
Horses, hornless unicorns, unicorns, pegasus, yaks, mammoths, bears, wolfs, lions, tigers, sphinxes, griffins, dragons, dinosaurs, gurhs, tortoises, terror birds, magic carpets, camels, locusts, scorpions, spiders, some machines, mobile goblin throne, dark beasts, corporeal beast and maybe even a jad too.