Honestly, I agree with him but his reasoning on it does suck.
Lack of parity in mobility is a huge problem unless it has a huge tradeoff. Think of some of the most annoyingly OP and the lack of counterplay champs that there are right now. Typically champs you'd consider Freelo.

What do they all have in common? Dashes, blinks, dodges, hops. In other words, overpowered mobility.
Why is mobility a problem? Because adds large amounts of both offensive and defensive power. Enemy low and running away? Gapclose and kill them. Losing an encounter, hop out. Trading at range? Hop into range, attack, hop back out before they can damage you. Skillshots? Sorry, they're pretty much nullified. Ganks? Sorry, I'm back under tower now.
Having higher mobility gives you the ability to control the lane. You can hop in to push them back, then hop back out. The main counter argument is to CC them. The problem with that is most CCs are either channeled/delayed abilities like
or are such short range you won't be able to target them
.
The key issue is, you have to distinguish mobility into two different types, movement speed and movement abilities. The problem comes in when you give high movement speed champs highly mobile abilities as well. The problem is Riot has not only given these champions high movement speed, and high mobility abilities, but high burst damage as well. This leads to a low risk, high reward dash in, fire your abilities dash out, giving you a very low risk gameplay style. While this is helpful against taking down adcs mid/late game, it also makes for an oppressive laning phase where it's very difficult to trade with your opponent.
For a better idea of some more healthy mobile champs see:
. While they are fairly mobile, they can't 100-0 you, dodge your skillshots/attacks, and hop back out before you even know what happened. They all have strategic weaknesses. The Freelo champs I listed above don't have that, short of
, which isn't really counterplay and has a 90s cooldown, longer than most of their blinks/dashes/hops and is only useful to AP champs.