How much did you talk about this with your doctor? Because questions like this are generally best suited to discuss with your doctor, who has a much better picture of your overall health and risk factors than we will!
However, I will note that the link between birth control and breast cancer is weak and poorly understood (ie, the study that observed the link didn't account for other potential risk factors, such as drinking alcohol). Harvard Medicine has a really good breakdown of what the study results mean: https://www.health.harvard.edu/womens-h ... ast-cancer
The increased risk for cervical cancer, too, is small, and can be more dependent on other factors (like contracting HPV) rather than the pill: https://www.mdanderson.org/publications ... 90624.html
And interestingly enough, the contraceptive pill can actually lower
the chance of contracting endometrial, ovarian, and
Taking the pill, much like anything else we do in life (the climate where we live, at what elevation we live, what we eat, how much we move, how much sleep we get, how much alcohol we imbibe, if we diet, how balanced our diet is, how well we clean our food for pesticides, how stressed we are, how often we clean our homes, etc.), is a choice associated with any number of risks that we may not even realize we are taking. The good news about the pill is that we do know quite a bit about taking it, and in this case, you can take a rather informed risk.
Yes, your grandmother counts for your family history, but one family member (and one further removed from you) shouldn't largely impact your chances of developing a disease/illness - and the study that found a slightly increased risk of breast cancer from the pill found that family history of cancer and being on the pill didn't seem to really impact the chance of breast cancer any further.
You're the one who can best make decisions regarding your own health, but I will say that taking the pill isn't what any reasonable person would likely describe as taking an "unreasonable risk" regarding your health.