Having spent a bit of time learning about functional programming, it's becoming more and more natural for me to want to work with static methods that don't perform any mutation.
Are there any reasons why I should curb this instinct?
The question I find a bit odd, because static methods and methods that perform no mutations are two orthogonal classifications of methods. You can have mutating static methods and nonmutating instance methods.
For me, it has become more and more natural to combine functional and oo programming; I like instance methods that perform no mutations. Functional programming is easier to understand because it discourages complex mutations; OO programming is easier to understand because the code and the data it operates on are close together. Why choose? Embrace the power of "and"; do both!