Here's my take on it:
Dragons are special. Sure there's lots of metagame reasons to have a unique dragon type, but a unique dragon type implies a special flavor treatment as well.
What is so special about them? Well, dragons are often presented as power incarnate (for example in the dragon shaman's description).
But in D&D there is no such thing as pure power. Now, what kind of power do they incarnate? Arcane sounds logical. They are the ancestors of sorcery and have vast arcane power. But sadly that doesn't link them to the non-arcane dragon shaman, nor does it link to some of their abilities, such as breath weapons.
What else is there? Divine/outer planar? No link, that's outsider/exemplars.
What about the inner planes? Many dragons have elemental subtypes, both true dragons and lesser versions. But elementals are the incarnation elemental planes (duh).
However, the elemental planes are not the only form of inner planes, and dragons are the most powerfull, iconic creatures that spring naturally from the material planes, and the material plane is made out of the inner planes.
So let's say dragons are the purest and most perfect examples of planar admixture. All planes channel into the essence of dragonkin in a way it is found nowhere else in the multiverse. The elemental planes infuse them with the elemental energies that brew in their organs, giving them their breath weapons and elemental link, the outer planes run in their vains and souls, giving various dragonkin their strong link to one alignment. The astral fills them with vast magical power. Lastly, the positive energy plane fills them with a incredibly long lifeforce, giving them their great age, vitallity and strength, while the negative energy plane fills them with destructive force and a truly frightfull presence.
This explains sorcerers as beings with a strong link to the astral plane, the source of magic. This may be through some connection with dragons or simmilarly connected creatures (naga, arnea, some outsiders) or by chance.
How does this explain the dragon shaman? Well, a creature of such powers not only combines it, it radiates power. The inner planar energies radiated by dragons can be snatchd up by creatures that have learned to do so through totemistic rituals. The dragon shaman forges a mystical link to one kind of dragon to snatch up those energies. When a dragon dies, his sould goes to the outer planes, leaving behind the powers of the elemental planes that suffused it, which will slowly seep into the essence of the material plane. But instead dragon shamans have learned to snatch up that power as well. Because the power they leech of is the same that fills dragons with their life force and elementalism, giving them a breath weapon, draconic likeness and the ability to channel positive energy (dragons can't channel their positive energy to heal other because they need it to maintain their vast bodies and keep the many different energies suffsing them in control). Like dragons they can radiate and unlike them even release their power, giving them a aura of power.
It's a bit wonky, but why shouldn't magic be vonky. Because they are power leeches and basically take power that should never have been theirs it's explained why a character with bad mental stats all around (thus a being of little magical connectio itself) can use these power easily.