There are four jobs that do, but honestly that isn't even that significant, except perhaps to Paladin. Using scaled GCDs instead of a fixed 1.5 second GCD regardless of Attack Speed, likewise, only affects three jobs. The main question there is, why not? There's no reason for Spell Speed and Skill Speed to be separate stats. There's no reason MCH, NIN, and MNK should receive zero benefit of Attack Speed on certain skills or under certain effects. It is less complex for Skill Speed and Spell Speed to simply be Speed, rather than each affecting only one type of skill, and we already know from the current Spell/Skill split that we can apply Attack Speed scalars to specific skills, allowing half-GCDs that actually can scale with Attack Speed, rather than the fixed 1.5-second ones.
The precedent for Skill Speed affecting more than just GCD speed has been set since mid-ARR when we did the exact same thing to DoT damage. Back then the discussion was whether to increase tick speed or increase tick damage. The devs called the prior impossible, and thus we have <Increases attack rate and increases periodic damage.> Later, auto-attack was changed from rate to damage probably as a sidelong nerf to NIN and PLD in preparation for SB, thus giving us <Increases GCD rate and increases auto-attack and periodic damage>. It's far from a one-stat-one-effect deal as is.
Leaving abilities unaffected means that either Speed must be balanced around high %oGCD jobs, which would make it ridiculously strong in the hands of low-oGCD% jobs, or left viable for low %oGCD jobs and nearly worthless for everyone else. There is no other way to balance Speed across jobs than to give it an ability component.
However, unlike DoTs, increasing the rate, rather than damage, of abilities, has actual potential harm in addition to its greater complexity, namely desync with other abilities and with raid (de)buffs. Thus, damage is all that's left. Thus: <Increases attack rate and ability and periodic damage> is your smoothest possible outcome (giving back AA rate in exchange for AA damage), and would scarcely be any more "needlessly complex/muddled" than Crit's "Increases chance to deal a critical hit and increases the damage modifier of critical hits." Both would at that point just be sensible, and, apart from their more exponential nature, quite balanced.
There are ways to make a DoT effective as a toolkit augmentation, but the means are badly limited so long as the margin of flexible GCDs between melee combos is so small. If I were to use a DoT on RDM, for instance, it would be a faint damage loss outside of AoE used in order to bank B/W mana. But this thread has yet to propose an example of using it to add flexibility or nuance, and nearly every DoT in this game (essentially, all but TC) -- at least in single-target combat -- works as if limited by those same warning signs you've mentioned, so expect people to expect the norm, which essentially means the worst. If and when that actual gameplay (beyond mere DoT maintenance) is given, well... the very quote you've quoted would suggest people would then be open to the idea, no?