You obviously have experience and insight here, I don't work for a council, so I will defer to your expertise. But, let me get this right:
It's claimed falsely by a huge number of people.
How do you know this? And if you do know who isn't being honest, then you'd be able to charge the correct amount for the household.
Is paid to anyone living alone even if they are a millionaire.
Yes, I understand this. But most single people are not millionaires. Far from it. And you're much less likely to be one if you live on your own.
Would be better spent on either reducing everyone's council tax
Right, so you're saying that as an adult who lives alone I should pay more (per person) than, say, a household of 4-5 people who use more services?
There is plenty of evidence to show that single/solo living people already pay more to live - food, transport, etc. than 'average families'. It's already unfair.