The Valdemaran Healers are astonished at the injury-healing - they can address that type of problem, of course, but it's slow and energy-intensive for both the Healer and patient, and patients who've sustained very severe injuries often don't make a complete recovery.
Their explanation of Valdemaran Healing is that it can only make the body do things that it would in theory be capable of anyway, the natural healing processes. It can boost the speed a lot, though, healing a wound in minutes rather than days, knitting a broken bone in days rather than months. More advanced Healing usually means doing this more precisely or selectively; a skilled Healer can coax a patient's bone marrow into producing more new blood, for example, without affecting anything else. There are some Healers who can regrow nerves, even a damaged spinal cord, which is something that usually wouldn't happen ever, though at a very small scale the processes it involves are natural ones.
The main advantage that Healers have for diseases is diagnosis; they can directly see what's wrong, where, at least with good training on how to interpret their Sight. For something like pneumonia, they can boost the body to repair damage and bring more resources to fight the infection. Some illnesses are still very hard to treat; cancers that have spread widely in the body, for example, are intractable for most Healers. Some rare diseases aren't really understood at all.