However, I read enough to discover how the deportation started, and I can understand why an official would have difficulty believing that a professional singer would be performing publicly for no comercial gain, if not for a charity gig.
There's really no need for the whole thing to be so damn Kafkaesque (shut in empty rooms, not told what's going on for hours), but in all likelihood that's more to do with the sheer incompetence of UK airport staff than any actual malice. And they really are incompetent.
I saw a fight break out in the passport line at heathrow airport one day, over a racist slur. I was amazed how far the staff let it go, they seemed compltely disinterested. In fact other civilians had to restrain the two fighters and the staff still did nothing until some of us started shouting "Aren't you going to get off your arses and bloody do anything?"
Useless bastards...
There are some other exceptions but most work or work-like activities you'd need a visa of some sort for.
The waiting tables example is "obvious", and while some activities are only compensated for a small elite (playing basketball, musical performance), there are a lot of activities that might be compensated and might not. Or be compensated indirectly.
For example, speaking at an academic or professional conference is "work", but it is probably not compensated (except possibly for a keynote or lead speaker). An author giving a book tour is work, but they are not getting paid for talks - instead, they may be "paid" in greater exposure for their work and possible increased sales.
London 101 though: It's tricky to see Big Ben as it's one of the bells housed inside the tower. The tower is officially called "Elizabeth Tower" (recently changed from the "Clock Tower" in honour of the Jubilee).
There are tours to see the bells (and Big Ben itself) but they seem to be only available to UK residents via their MPs. I've never bothered to try (I'd rather sort out a tour of Lord's cricket ground first).
I've realised a good analogy would be if a golfer was going somewhere on holidays and was going to play a few rounds of leisurely golf with friends while he was there.
The thing with the woman officer who stopped my friend and then interviewed her was that woman lied saying that she phoned music venues and they told her how many thousands of pounds my friend was going to be making. Completely. Ridiculously. Fabricated. Just started making stuff up to back up her case.
That said, it's terrible for the typical HN purpose, which is to extract maximum information in minimum time. I skimmed a bit, said, "Wha?" and moved on.