Couldn't the races then just be less (at least universally) stat-based? The game already runs largely on conditionals, now that our stats are separate for every class we level.
For example, there's no reason why there couldn't be a benefit for any given role that still functions differently than other races' benefit for that same role.
Consider Tanking:
Taru - Stubborn Persistence (Effective hp bonus especially when health is low [Taru-size]),
Hyur - Bastion (increased defense from surrounding non-offensive (especially party) effects),
Roegadyn - Avatar (increased defense and health regeneration from surrounding offensive and buffing effects)
etc.
It would take a little work, but I'm sure racial benefits like that could be made for all roles (or classes or main-stats). The question after that is simply whether or not the benefits under every condition for whatever race should follow the same general specialty ("each bonus sorta points towards playing a caster," or "towards an intelligence stat"), or, inversely, should not stack and remain separate.
(That said, for a race to have a general move towards a certain stats wouldn't really limit freedom until we have an actual choice in stats -- at present, the ones useful to us per class are pretty well pre-defined. It wouldn't be such a bad thing for Roegadyn, for example, to make a better use of strength stat even when playing Conjurer, allowing them to be a more capable Shield-Wand tank if they like, but not especially gimping them compared to other races when going for full dps either.)
In the latter case, races could still pull off any role to the same extent, just through different strategy (the bonuses on the whole do not force a particular role). A party's racial makeup would be a fairly important factor, but not in a WoW-ish way of "do you have the right gem slotted?" (the right race for the role), but rather a gestalt of "do all these factors together make us team badass?"
----I still favor the idea of traits-leveling, where these racial benefits would probably be traits you could level from the beginning, where others would be learned through your leveling experience. Their significance too then, would be a player-controlled factor.
@ Reinheart
As for racial animation differences, I'd like to see more animations, but probably not have them based quite on race. Leading stats (or my vaguely imagined version of traits) seem like they'd be a more interesting reason for differentiating animations.