This is very true.
Mobility is more important in this game than any other metric. Second is attack speed (including endlag on attacks), third is size, and fourth is KO power. A character can have garbage damage, light weight, iffy recovery, and mediocre range and still be top tier, or even the top of top tier, with some good mobility parameters, either attack speed or the ability to net KOs (you don't need both), and a decently sized hurtbox (as in, not Bowser-level, but smaller characters still have a general advantage over larger characters, just look at ZSS vs. Pikachu). We have living proof of this in Mario.
When we say "mobility," though, we're actually talking about more than one thing. The reason that mobility is so strong is because it's an umbrella term for several stats:
- horizontal aerial acceleration and speed
- vertical aerial acceleration and speed
- jump height
- initial dash speed
- walk speed
- run speed (least important)
When a character has good stats in the above parameters, he seems to "flow" really well, and high-level players will gravitate to him. With good mobility, you can:
- Get out of bad aerial situations by drifting away. Corollary: if you're not a fast faller, you don't get combo'd as much
- Follow up on your attacks more easily both on the ground and in the air
- Be more unpredictable in general just by moving around; also, "feint" attacks, by moving in and out
- Abuse your opponent's misspacing
- Space yourself in such a way that commitment-based characters with good range are uncomfortable throwing out their attacks
- Take stage control more consistently and easily in neutral
All of these are huge. A character with good mobility can apply his mobility advantage to advantage, disadvantage, and neutral. It's not a strictly one-dimensional advantage like weight. Range is similar in that it applies to all the states, but not as blatantly as mobility (like really, how much does your range matter in disadvantage, for example? you can swat people away, but we all know how good Shulk with his 11-frame n-air is at doing that! he'd prefer to be able to jump/drift away, which he does get to do in Speed/Jump).