I think that Korra could heal using water-bending, but what about blood-bending, etc?
As far as I know we have never seen an Avatar bend blood, metal or lightning. As you say, Korra knows how to heal with waterbending, but that isn't a very rare ability.
It is not known exactly how Avatar Aang was able to overcome Yakone bloodbending him, but it is possible he overpowered him by bloodbending himself while in the Avatar state.
I think it's worth mentioning that these other "sub-techniques" aren't really that different, they just require more skill. Bloodbending is bending water in the blood, metalbending is bending the earth impurities in metal (hence why it doesn't work on pure metal) and lightning is a very powerful form of fire.
The Avatar has to learn all four bending disciplines, plus all the spiritual jazz, and also travel the world solving problems, so he or she doesn't have the time to devote to learning these extremely difficult techniques. But there's no reason to think that he or she couldn't do with enough training.
No Avatar has been shown using the advanced techniques of any element that is not the element they were born into, with two notable exceptions. Korra was born to waterbenders and is the only Avatar known to be able to heal with water. As gentle and caring as Aang was, the fact that he was never shown to be able to heal is rather telling, to me. Likewise, Korra should likely have learned to use lightning, given her strong affinity with fire despite her waterbender birth.
Notable exceptions to using advanced techniques: Aang did "see with his feet" using Earthbending against Phoenix King Ozai. Presumably, that can count as "a niche use of earthbending" any Avatar can learn if they're inclined and not an advanced ability, since Toph invented it at a very early age and didn't create metalbending until she was, according to herself, the greatest earthbender in the world.
Number #2 can also be justified, but it's a stretch. Aang also redirects lightning during that same battle. He never creates lightning, but he does redirect a bolt. Perhaps, again, you can call it a niche use that you don't need to learn to create lightning to do (I'm not even certain you see Zuko ever throw lightning, even though he also learns to redirect it), but it might be a little harder to see.