In combat, the vampire is too busy for an automatic hit
The bite attack describes as valid targets "one willing creature, or a creature that is grappled by the vampire, incapacitated, or restrained." A creature who is not any of these things cannot be attacked with a bite. However, merely having a target who is willing is not sufficient to make the attack automatically successful, any more than having a target who is unable to move (such as when restrained or incapacitated). So why not?
What is the situation? Is it not a conflict...
The vampire has slipped into a willing victim's bedchamber. The victim presents themselves, and the vampire initiates the draining bite. Either there is no one else present, or those who are present are unable to do anything but watch on in horror - they are held, or too far away, are scrying the scene but not physically present, etc. This is not a conflict, and the vampire should be permitted an automatically successful bite.
The DMG Section on Using Ability Scores (p.237) says:
When a player wants to do something, it's often appropriate to let the attempt succeed without a roll or a reference to the character's ability scores...Only call for a roll if there is a meaningful consequence for failure. When deciding whether to use a roll, ask yourself two questions: Is a task so easy and so free of conflict and stress that there should be no chance of failure?
Here, the vampire has all the time in the world, and no distractions that will prevent them from draining their victim. An attack roll is unnecessary, if this scene is even 'on screen'. It is a similar situation if the victim is unconscious and alone - absent any combatants, we can indeed assume automatic success.
Or is it combat?
Suppose you have the same vampire and willing victim, but now there are other characters on the scene - in particular, party members who want to save their charmed ally. The vampire can attempt a bite on a willing creature, but this is in the middle of combat:
The clatter of a sword striking against a shield. The terrible rending sound as monstrous claws tear through armor. A brilliant flash of light as a ball of flame blossoms from a wizard’s spell. The sharp tang of blood in the air, cutting through the stench of vile monsters. Roars of fury, shouts of triumph, cries of pain. Combat in D&D can be chaotic, deadly, and thrilling.
The vampire cannot focus its full attention on the victim, willing or not.
In combat, most creatures stay alert for signs of danger all around
In addition to biting the victim, the vampire is simultaneously defending itself against the PCs, perhaps even taking and regenerating damage. It is commanding its minions. It is alert for the presence of anyone trying to sneak up on it. It is evaluating whether or not to use its Legendary Resistance in response to a failed saving throw.
Here, the attack roll does not represent so much the chance that the vampire is able to bite a willing victim, but the chance that it is able to do so uninterrupted from everything else that is happening within the chaos of combat going on around it.
Narratively, you could consider the vampire as leaning in close, scenting the warm blood of its victim, the heat of their living flesh so freely given - and then someone drops a sunburst spell at the last second, interrupting their bite, just when they were so tantalizingly close.
It is the DM's prerogative to assign advantage as a circumstantial bonus - and certainly advantage on the bite's attack roll would be appropriate for a willing victim, just as this advantage is automatic on an unconscious foe.