I'm someone who often doesn't use boots, but I can tell you this: WIthout boots, Draven's slow/displacement isn't enough to escape. WIth boots, it is.
The same applies for Zoe. If she has boots, her sleep will allow her to get away.
But there's a far more important reason to have boots:
To be exposed to more fights. Having ~ 400 move speed in a 500 move speed game is VERY bad, because every time you recall, your team will be prone to start fights before you arrive, and you will get to cs waves slower.
The main exceptions to the move speed rules are champions like
or
who add flat amounts of move speed to themselves (and possibly others) in large amounts. However, even
is prone to throwing a game without boots, because often, her E isn't enough to get away if her W is down, and she's recently been "in combat." With boots, her slow is enough to get away to start proccing her W again.
on the other hand has a dominant cc presence, that can stun chasers, and her best builds give her and her allies +120-150 base move speed every time she self skills. Coupled with items like
, she can pretty much get intermittent 600-700 move speed wihout shoes.
You can do the same with champs like
or
, but the key difference is: They give % based move speed, so adding boots to the equation make them even faster by raising the base move speed, so if you don't want to start a game with 0/6
just to provide her agency.
I even get boots sometimes on
, simply because the other team is going to be an AP nuke team with CC, and I need
. Or if they are assassins based on auto attacks like
,
helps a ton at keeping my deaths low and possibly wasting their ults.
FYI: I average 61% Kill participation rates. That means.... generally I am exceptional about getting around the map and into the fights all over it. Sometimes that's because of things like 
(Boots are the only defense to a
btw)