The original concern was
> Are you comfortable with your bank knowing who you hand cash to?
You moved the goalposts to money-launderers, but let's assume only money-launderers, which I am not, because it is fun thinking about these things.
Your hypothesis only holds if at least some pubs, supermarkets, record serial numbers, or if we assume that "money launderers" always pay in 200 or 300 pounds and there is an active investigation of cctv cameras.
Now, going back to the original question:
> Are you comfortable with your bank knowing who you hand cash to?
This is not a concern at all, because the bank does not know who I hand my cash to. Cash that the bank recorded as given to me can travel many many times before the bank gets to see it again, thus proving nothing about who I handed to it.