It is a cultural thing.
I come from a country where bar tenders pour spirits directly into the cocktail, without a measure. They work for tips so they're keen to be generous and make "strong" drinks. If a bartender in that country were to pull out one of those little metal measures and meticulously pour the precise amount into the glass, the customer would think he was being stingy. It would create a bad feeling.
But, really, in British culture, it's about fairness. The customer wants to know that they are getting exactly what they've paid for. The law (Weights and Measures Act) backs them up and if a bartender just poured straight from the bottle to the glass, they would feel uneasy. They would be worried that they might be cheated. Or perhaps their drink would be "too strong."It would create a bad feeling.
It's not stinginess - it's a rigid adherence to a sense of fairness.
I don't think it's difficult to have a tenner lying about (away from the real stash) in order to make change. It will make for a better vibe during the booking as the client feels that everything is above board and fair.