Hmm, the majority votes were about what I expected.
> DPS with a focus on utility/support.
Was tempted to put "Not listed" just because I think I might be imagining that differently than most though, if any previous combos are an indication. I'd like the Dancer to be a sort of combo-middleman or small scale battlefield-manipulator, with about the damage of a (skilled) Bard, and a hefty raid dps contribution capable of re-centering group tactics a bit around what all it provides. I'm not remotely interested in the various "dances" and would prefer that those effects be dealt instead through the actual movement and choice of attacks the Dancer performs.
While any system can potentially work, that assumes larger surrounding changes to clear those particular obstacles. It may well be that come 4.0, this comparison between TP and MP won't even be a thing. One or the other may be removed entirely, jobs may each take on unique ways of regenerating TP, raid support for either may be increased or decreased in weight or dependence, every class may end up with a means of MP consumption, so on and so forth.
But, assuming the present systems remain for the time being, those particular obstacles could cripple the idea of a TP-based healer. Moreover, they could cripple the Dancer's means of differentiating itself from the others. What difference does it make if it uses TP instead of MP, but consumes TP only at the same rate over time from both attacks and heals that any other healer would with MP, and regenerates the same proportion over time with Invigorate or its own skill that other healers generate with Shroud, LA, or AF? None, except that the Dancer is most likely in melee range and is hopefully given the mobility to compensate for that.
Sprint at present would cripple a TP-based Dancer. Assuming the idea is even considered, so too would likely be changes to Sprint, negating its TP cost, making it a toggle, or even uniquely having Dancer consume some amount of MP for mobility, including Sprint, instead of TP. But at present the Dancer would be locked out of its use, relying only on its unique mobility skills.
The lack of Mage's Ballad and Promoted Bishop are less significant. Rather than a lost resource, it creates a resource split between the Dancer and other healer that would otherwise be present only in long or AoE-required fights with physical dps, and ultimately creates a meta similar to MCH's Hypercharge, where either all casters or all physical are preferred for maximum effect. Dancers, being physical, would be the go-to off-healer in an all-physical composition, while deweighted in compositions that are caster-heavy. Nothing new there, apart from the NIN synergy, which we already see between Ninjas and Skill Speed classes and Warriors.
This could actually be just as crippling. Unless the Dancer is MP-efficient enough (similarly to a Cure/Physic/Benefic spam and HoTs only full time healer) not to require time spent attacking, it would necessarily have downtime from healing, and would be unable to regenerate during actual fight downtime. This could be immensely unbalancing; if the Dancer is balanced for the downtime inefficiency, it will over-perform in fights without it, while still likely under-performing in fights with it. Due to the homogeneity of our healers so far, that would stick out, badly. If MP could instead be regenerated also via non-targeted weaponskills (e.g. "dances") then this would be less of an issue.
While such a healer would likely excel in the current meta, this could be problematic in entering a new tier of raid content, especially if healing requirements came to be tuned higher. That issue can then be further mitigated through role-specific, rather than job-specific gear, and so on, as a Dancer main to-be could temporarily be something else until geared enough to make the swap to the then more efficient healer, whose healing downtime is still primarily DPS uptime.