In general, why are there so few flights to London City Airport?

LCY (London City) is a small airport with a short runway. It is built in the middle of the former docks with water on both sides and at the end of the runway, so the runway cannot be extended. Look at the photo here.
Because of the short runway it can only handle small short take-off and landing aircaft. Because of the small aircraft ticket prices are higher, it is mostly business-orientated destinations in Western Europe with customers travelling to central London and willing to pay for the time saving in not having to travel in from Heathrow or Gatwick.

P.S. Have you looked into flights to Stansted? That airport has good connections to East London.

There is one flight from New York to LCY - BAs all business class flight on a baby Airbus A318.

There are several domestic routes, and many European, the destinations often having a 'financial' flavour to them; Switzerland, Luxembourg, Antwerp etc.

As the other two have said, it serves smaller planes and generally western Europe. My Air France flight from Dublin landed there and it was terribly convenient. It was quick to get out of, easy to get to the tube and close to so much. I think I would pay extra in the future to land there.

I would warn people, however, that it is very inconvenient to connect between City and any of the other London airports. Some people on the board in the past have been unpleasantly shocked by this.

I would widen out the above to be inconvenient to change between any of the various 'London' airports even though there are different degrees of it. Even on a single ticket where you are protected against misconnects I wouldn't recommend it.

About the only 2 London area airports reasonably close to each other as the crow flies (or I suppose the plane flies) are Stansted and Luton. A taxi would do the job. But it would be a very strange itinerary which would have a Stansted/Luton connection.

Stansted to Luton would be a very expensive taxi. The only other ground transport between the two is a slow and infrequent bus which is often late, or 'V' shaped trip by train into London and out again.

I would also advise against changing at Heathrow if it is different terminals. Terminal 1-2-3 are OK, they connect with walkways both landside and airside. Terminal 4 is on the south-side of the airport, and Terminal 5 in the west. They are effictively totally different airports.

There was a recent post about using Birmingham airport for the Cotswolds. For the West Country, including Cornwall, both Birmingham and Manchester can be quicker and easier to get to than the London airports.

Now we are really off-topic!