But why does the AFM prohibit it?
The short answer (without getting into a huge dissertation about aircraft certification regulations) is that the wing tanks were not tested during the take-off, climb, descent, and landing phases of flight. This was a logical decision by de Havilland (back in the day), because wing tanks are only needed during cruise - they exist simply to prolong the range of the aircraft.
The regulators who are responsible for aircraft certification (in the case of Canada, this is Transport Canada) are very safety-conscious people - something that all of us are very grateful for. So, when the regulators are reviewing a proposal from an aircraft manufacturer for a system such as wing tanks, the first thing they ask for is flight test data that proves that the system is absolutely safe and reliable under all possible circumstances. In the case of cruise flight, this means making sure that the wing tanks will work in all different aircraft attitudes, within all the permitted loading configurations, within the approved 'g' envelope, and so forth.
When the wing tanks were first submitted for approval by DH back in the late 1960s / early 1970s, DH only substantiated safe operation of them during the cruise phase of flight. This means that a limitation was published stating that they are only approved for use during the cruise phase of flight.
I can give you a similar, more recent example: The new Series 400 Twin Otters have two cigarette-lighter type plugs in the flight compartment for charging up cell phones and stuff like that. We (Viking) did not carry out 'flight tests' of these power outlets. For that reason, there is a placard near each outlet (and a limitation in the AFM) that says that the power outlets may only be used when the aircraft is on the ground. This is because we only tested them on the ground, not because anyone is aware of any possible problems that might arise if they were used in flight.
I know it all sounds very restrictive, but if you think about it, it is all directed towards operational safety. It's a good thing.