I'm not the one moving though. The game says your speed is halved, but it doesnt say anything about a mount you ride.
No, it doesn't say specifically. They *don't claim* to give rules specifically for everything, and that does not mean that anything not specifically stated is automatically okay. This is "rulings, not rules" stuff.
It isn't like you are on a huge airship, that can be considered large enough to have a fight on. This is a flying mount that can perhaps carry two, but normally only if they are cooperating. Your target is not cooperating. That's really should give your mount issues.
In the PHB, the warhorse, who has the same strength as a pegasus, is given a carrying capacity of 540 pounds, that *should* be enough for one other creature
Don't count on it. A nice, strong, human warrior is probably a 6' tall, 200 lbs slab of muscle, wrapped in 50 lbs of armor. Two of them puts the steed at 500 lbs. The exotic saddle needed to ride a pegasus is anther 40 lbs.
I hope you and your target are both skinny elves in leather.
And, as to why the grappling half-speed thing should apply to the mount: A Clydesdale horse (our stand-in for a warhorse) comes in weighing about a ton. We just talked about having 500 lbs of warriors on its back. That's *a quarter of the animal's weight*, *struggling*, on its back - that grappled person is't just relaxing down there, you know. The animal is running close to its capacity, even if it isn't over. This is not a reasonable state in which to expect the animal to move at full speed.
Or, you know, fine. The mount's speed is not halved, but yours is. You now have a fly speed half of that of your mount. This means the mount flies faster than you, and your fall off! Fun!
And, the point about needing to shove (which requires an attack) is relevant. Once you are up in the air, the target is no longer struggling to break free - they are struggling to hold onto the straps and all of your saddle to not fall.
All this boils down to - yes you can grapple, fly up, and drop the target. Once. Not twice. Twice would be a little much.