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RAW do not show the need for a free hand to ‘hold on’ to the mount when making a DC for cover.
I can’t find any clarification on this, and while it would make sense to need a free hand I don’t want to presuppose rules rational that a skilled rider could do it with an arm slung around neck of mount while holding a weapon or shield.
There is no requirement that you have a free hand to take cover using Ride.
As for making sense, Pathfinder rules do not need to make sense. There are many instances where they make no 'real world' sense whatsoever.
Example: a 200lb level 1 character can have 300lbs of dead-weight (no buoyancy) on his person and still easily swim in calm water and easily tread water in rough water for 2 hours or more!
Swim skill = 1rank, +3class skill, +4strength, -6 ACP = +2
Calm Water DC: 10 (take 10)
Rough Water DC: 15 (take 10; treading water only requires a 10)
The first fatigue check is after 1 hour, even if you fail it is only 1d6 nonlethal. It will take 2 hours minimum and an average of 3 hours for you to suffer enough nonlethal damage to knock a first level martial character with 10hp unconscious.
Funny bit, this would've been impossible in 3.0 (-1 ACP per 5lbs) and in 3.5 it would've been extremely difficult to do this in rough water (double ACP penalty to swim).
Clearly a trend of simplifying rules at the cost 'realism'.