Is the method at all effective? Can eyesight truly be improved/repaired through exercise and lifestyle changes?
Unsurprisingly, it does not work.
As any other "alternative" medicine, it should be treated with the utmost caution, the general principle being "alternative medicine that works is called medicine."
The findings of Visual Training for Refractive Errors CTA - October 2004, a meta-analysis from the American Academy of Ophthalmology, are summarised below by Wikipedia (emphasis mine):
No evidence was found that [visual training] techniques could objectively benefit eyesight, though some studies noted changes, both positive and negative, in the visual acuity of nearsighted subjects as measured by a Snellen chart. In some cases noted improvements were maintained at subsequent follow-ups. However, these results were not seen as actual reversals of nearsightedness, and were attributed instead to factors such as "improvements in interpreting blurred images, changes in mood or motivation, creation of an artificial contact lens by tear film changes, or a pinhole effect from miosis of the pupil."
Wikipedia goes on to quote a second report:
In 2005 the Ophthalmology Department of New Zealand's Christchurch Hospital published a review of forty-three studies regarding the use of eye exercises. They found that "As yet there is no clear scientific evidence published in the mainstream literature supporting the use of eye exercises" to improve visual acuity, and concluded that "their use therefore remains controversial."
The Bates Method relies on the idea that the eye focuses via accommodation using muscles which change the eye's shape. The accepted idea that the lens is primarily responsible for focus, and that the shape of the eye doesn't change much, if at all, to focus wouldn't be difficult to verify objectively. The idea that science might try to suppress the truth about this is merely a conspiracy theory.
Dr Phillip Pollack explains on QuackWatch:
It would be theoretically impossible for the extrinsic muscles to alter the structure of the eyeball so as to meet the requirements of accommodation. The outside, white coat of the eyeball (the sclera) is not resilient and elastic, as shown by tests in the laboratory. Furthermore, when pressure inside an eye is increased by more than 500 per cent, the volume of the eyeball hardly changes, as shown by measurements (the increase is only 0.007 per cent of the original volume). This proves that the sclera does not yield very easily to pressure. Finally, the sclera becomes even more rigid and less resilient with age, especially after the age of 40.