Can chartering a private plane be an affordable alternative to scheduled flights?

I wouldn't call it an alternative, but on some smaller airfields you might be able to score a cheap flight if you call them up 1-2 weeks in advance and organize your trip. Pilots are required to do a certain number of flight hours each year to keep their license, and they are usually quite happy to have someone on board to offset the costs a bit.

However, if you book a small 2 or 4 seater they might cancel or delay the trip in the last minute because of bad weather. Your luggage space and allowance is very limited, depending on the airplane. They also may not want to land or takeoff on big international airports because of high fees.

In short, chartering planes is almost never cheaper than flying commercially. Even in Andra's case, chartering was only "cheaper" because they were apparently in a blazing hurry; if waiting around for a day or two had been an option, they could almost certainly have found a cheaper option, or made the 700 km trek by bus or something.

I once looked fairly seriously into chartering a plane for about a hundred people. The basic problem turned out to be that, while the actual flight itself at $5000-10000/hr would have been doable at around $400 per person (2 hours each way x 2 / 100 people), keeping the airplane and the pilot on the ground in a destination where they can't do anything (= make money) for several days is very expensive, since the waiting price is not much cheaper than the flying price. And if they fly home and come back later, you're now looking at paying for two roundtrips plus turnaround time, meaning the total cost is now around $1000 per person.

By contrast, commercial airlines are very good at keeping their planes in the air making money (some LCCs approach 20 hours out of 24!), so they can amortize their fuel, crew, insurance etc costs across lots more passengers, and thus offer cheaper fares.

A new service called Airpooler lets you pay the price of the fuel, since it's not an officially licensed charter service, you can't be charged more, and are essentially 'hitching'. It's small scale for now, but may well grow.