Hell no, governments shouldn't be run by polls. They should be run by people with strong moral compasses that will stand by the principles of right, and avoid the wrongs that they might otherwise inflict upon their citizens or other persons throughout the world.
Running governments by running up trial balloons, or sticking a finger up into the wind to gauge which way the winds are blowing leads to governments of weakness, and leads to making choices that can come back to haunt the people that have to live with them in the future.
Thanks but no thanks. Govern from the center sure, but govern from polls, no friggin' way.