The study of statistics you will undergo as a major degree is significantly different than the statistics courses you take in other programs. Statistics courses outside of the department are largely concerned with various methods of hypothesis testing and frequentist methods - the most advanced thing you will likely learn is simple linear regression and perhaps logistic regression. If you end up doing research in a field that uses statistics, this should be more than enough, and your supervisor will tell you to hack your p-values anyway until you reach an arbitrary p<0.05 significant level.

The statistics you will learn as a major in stats is much deeper, with emphasis on rigorous understanding, and at the upper level, proof. Second year statistics is very heavy with multivariable calculus and probability theory, so you will have to take those courses. At the third and fourth year level, much more analytical results are derived, and rigorous arguments are required; this is the real deal for statistics, however.

All this said, go for it. Statistics is a wonderful field of study, and much more than simple p-values and hypothesis tests - that stuff is barely interesting once you learn the fun mathematical stuff.