No, that isn't the case in most vampire stories, I could have sworn that it is the vampire's blood that turns people, but the vampire has to drain the person of most of their blood, and then they have the person drink some of their blood to turn them.
I think they should keep the current vampires the same, but I agree the Volkihar should be added, and the way you said they should function is how I imagined they would be. Vampire quests are indeed needed, and they should have a quest that involves Lamae Beolfag.
What do you mean about parallels in Skyrim, and the previous games? I understand the Three houses in Morrowind, and the three clans of vampires, but I don't understand the other parallels.
In previous games, and in Skyrim itself, the prominent Vampire clans would mimic, in structure and style, the mortal civiliation in which they are surrounded. This is an interesting gameplay design by the Devs, not Vampires trying to fit in. For Example in The Elder Scrolls III: Morrowind, there were three major Dunmer Houses on the island. Houses Hlaalu (skiled theives), Redoran (skilled warriors) and Telvanni (powerful wizards). The three prominent Vampire clans on the island were Berne (thieves), Quarra (warriors) and Aundae (wizards). You can do the research and find more in common between the groups. In The Elder Scrolls: Oblivion, the Vampires were similar to the Imperials because of their common history of conquest and that they both happened to have pacts with gods, respectively Akatosh and Clavicus Vile. In Skyrim, as there is a civil war going on between the Nords and their Imperial allies, the Vampires from Cyrodiil have apparently been trying to take over the prominent Skyrim Vampire clans, most notoriously the Volkihar. This pattern is seen throughout the series where Vampires are featured.
And I completely disagree. Vampires in their current form in Skyrim are incredibly boring to play and need a massive revamping, as do Werewolves (and Werebears).