I don’t think that it’s always a case of police not investigating when there is just one complaint, I think it’s more a case of the difficulty proving it.
Abuse is rarely witnessed by a third party, the only witnesses are the abuser and the victim. If there are no physical signs of abuse and no witnesses and no cctv, what do the police have to go on other than an accusation? If there is no previous history and no previous cause for concern and the accused denies it, where do the police go from there? Someone making an accusation is not enough evidence in itself, there has to be more.
That is why abusers get away with it for so long.