Yes, they do
In the rules, mundane things work as we would expect them to from the real world, so there is no need to spell out what it means to mount a horse, because it works as expected from the real world: you are sitting on the horse. And by extention, as shown in the rules snippet you quote, the same holds for phantastic creatures like pegasi or griffons.
The benefits to "speed and mobility" you gain are those of being able to use the mount's speed and modes of movement, instead of your own, which in many cases are better than yours. You as a typcial character have 25 or 30 feet of walking movement, while a riding horse has 60 feet, twice as fast, and a griffon for example has 80 feet of flying movement.
There even is additonal rules text supporting this, even though there is no real need for stating the obvious. The rules for mounted combat (p. 198, PHB) say:
In either case, if the mount provokes an opportunity attack while you’re on it, the attacker can target you or the mount.
So it is clear what your relative position to the mount is that you are on the mount.
The DMG (p. 119) for overland travel also explains:
Flying by spell or magic item works the same as travel on foot, as described in the Player's Handbook. A creature that serves as a flying mount must rest 1 hour for every 3 hours it flies, and it can't fly for more than 9 hours per day. Thus, characters mounted on griffons (which have a flying speed of 80 feet) can travel at 8 miles per hour, covering 72 miles over 9 hours with two I-hour-long rests over the course of the day.
This makes it explicit that when you are using a flying mount, you use the mounts speed, instead of your own.