To cover the full spectrum, you need something that acts like a blackbody radiator (e.g. hot element in tungsten), rather than fluorescent, as you mention, since that has sharp spikes in its spectrum (as do xenon) (though there are fluorescent panel-style lighting solutions).
See examples of emission spectra
You also don't want LED, because the emission spectrum is narrow.
I'm not sure how even color temperature across a beam will vary. If the diffuser or light modifier is constructed well, there should be full symmetry in the distribution of frequencies across the beam.
A major issue that should inform your decision is how large you want the light source to be. How large is your scene?
Anything can be made a point source, if you move it far enough away from the subject, but some lights cannot be made bright enough to do this. A bright incandescent bulb is excellent for closeup portrait work (left shot in this panel lit by 60W bulb).
HMI is very expensive, but the filament is tiny and shadows are extremely sharp from this kind of lighting (it is also much closer to daylight than tungsten, in addition to being available in crazy-high power versions, which require transformers) (right panel in this shot lit by 1200W HMI). HMI gives a brilliant, near-daylight light.
Don't dismiss fluorescent out of hand. Some of the cheapest and most interesting light sources are fluorescent. For this shot, I used a round (~1.5 ft diameter) fluorescent bathroom light, and shot through it. When shot with 50/1.2 and rendered in b/w the airy quality is wonderful. Here the background was blown out with a couple of 500 Ws monoheads in the background.
If you want to diffuse the light (softbox, scrim, etc) you have to consider the temperature of the source. You can't use the same modifiers for continuous light as for flash work - they would melt.
In other words, a lot about the light source has to do with how it's packaged and shaped.