There's a good reason for us to be unsympathetic to the Doctor. Each invasion of Earth and other major crisis is his fault. I've read enough time travel stories to deduce the fundamental principle of time travel: The Conservation of Probability. In equation form:
If some highly improbable (P1) event with a good effect (E1) happens an equally improbable event (P2) with a bad effect (E2) must also occur to maintain the balance of the universe. They don't have to precisely match as long as the signs are opposite and the total comes to zero. Some invader would cause lots of damage compared to the benefits the Doctor would provide by hanging around London, but he's also much more improbable, so it balances out. In fairness to the Doctor, all those times the TARDIS gets yanked off to some unexpected time and place is probably him being used to balance out some improbable bit of evil.
Note that this works in different dimensions than good/evil. If slobby time lord the Bachelor visited New York leaving a trail of pizza crusts and candy wrappers, an alien robot with a tropism for styrofoam would arrive and consume an equal and opposite volume of trash.
Potential time travelers in my audience should beware. If you travel back in time and provide a huge knowledge boost in electronics and cybernetics, someone else will come along and trash the school system so potential scientists will be reduced to illiterate drones. Oops, too late. Try not to do it again. The path of maximum probability has us making good progress and the best way to improve is to have the highly-probable people already here work on it.