Rules as Written, yes
What is a 'willing target?
As quoted in the original post (emphasis mine)
Although the target isn’t under the vampire’s control...it is a willing target for the vampire’s bite attack.
By specifically designating the charmed target as 'willing', the text is differentiating it from other targets. There is no fluff; the words have to mean something.
In this particular case, the consequence of being a willing target for the vampire's bite refers back to the text in the description of the bite itself:
Bite (Bat or Vampire Form Only). Melee Weapon Attack: +9 to hit, reach 5 ft., one willing creature, or a creature that is grappled by the vampire, incapacitated, or restrained.
A vampire cannot normally make a bite attack against a creature; to be an eligible target, a creature must be grappled (by the vampire), incapacitated, or restrained. Or willing. In this case the effect of the charm is simply to allow the victim to be a target of the bite without also being under one of the other three conditions.
When we consider each of the other steps that accompany the bite attack, we can see that RAW being 'willing' does not affect the mechanics of any of them. When the bite attack is made, the vampire makes an attack roll; if the attack results in a hit, damage is done and the target's hp maximum is reduced, and finally, as a result of the damage, a save is triggered.
Does being a 'willing target' allow attacks to automatically hit? No, it does not.
Does being a 'willing target' allow the save against the charm to be automatically failed? No, it does not.
With no other consequence of the bite attack being affected by the target's will, and with the targeting of the bite itself specifically being dependent on the target's will, it is clear that it is the targeting only to which the 'willing' refers.
Experience in Actual Play
In my experience as a DM, events on-screen should be rolled out as RAW; off-screen events should serve the narrative. This advice follows from the general principle that the game is about the PC's choosing actions, not the DM making rolls.
If the PC's are present when the vampire is biting the victim, whether or not the victim makes the forced charm save will likely affect their next actions, particularly if the victim is a PC or an allied NPC. In this case the save should be rolled out, as RAW.
However, if the PC's are not present, the DM should feel free to simply decide whether or not the save is made as it better serves the story. What you absolutely do not want to do is make the players wait while the DM resolves events RAW by making a bunch of rolls that don't immediately impact them. For example, suppose that Ireena has been captured and charmed by Strahd but the PCs are on their way to the castle. If it furthers the story to have Ireena make her save, escape, and meet them outside, it is better to just assume that it happens; if it furthers the story to have her fail and still be charmed when they arrive, then just decide that is what happens. What you don't want is a bunch of players waiting on you while you resolve things RAW that don't directly involve them:
"Ok, Strahd attempts to bite Ireena (roll), succeeds (damage roll), prompting a charm save (roll), which she makes! She tries to pass herself off as still charmed (Deception vs. Insight, roll, roll) and fools Strahd! Ok, she waits until he leaves and tries to sneak out of the castle...(Stealth vs. Gargoyle Passive perception, roll, success), (Stealth vs. Dragon Passive Perception, roll, fail), oh, but she is discovered and recaptured just before she reaches the door. Nevermind then, where were we?
