Yes.
But what likely happened in the case that PP spoke of is that the Police didn't say 'do you want to press charges?' but said something along the lines of "we think a criminal offence occurred here, would you like to make a statement in this case to support a conviction and would be willing to testify to that in court?".
Most people interpret that as 'do you want to press charges?' because that's a very common in-use term even if not accurate legally. But the above conversation is usually what they're referring to.
(Same as thousands of people in the UK, including many on MN think there is a charge called 'statutory rape' i.e an older person having sex with a child under the age of consent when there is no such offence in UK law but it's common terminology and assumption)
And if the alleged victim says no so saying without saying it "I wouldn't like to make a statement in this case, support a conviction and testify to it on court" then the Police decide if it's serious enough and in the public interest to proceed to an investigation without a victim statement and then decide if there is enough evidence to present to the CPS.
They might do or they might not depending on what evidence they have.
Then the CPS decides to proceed to charge or not. And part of that decision is whether there is a realistic prospect of prosecution.
Without a victim or witness willing to make a statement to the offence and appear in court, the CPS can conclude that if the alleged perpetrator does not plead guilty, or pleads guilty but challenges some of the evidence, it goes to court and the defence team could potentially rip the case apart very quickly, depending on the other evidence available.
So yes, the Police can proceed with an investigation without a victim supporting it but there are many factors involved that contribute to whether that file is passed to the CPS or not and whether the CPS would proceed with a charge.