one-sample z-test for proportions

hello, I wanted to know why the one-sample z-test for proportions is used instead of just using the binomial formula to find p-vales? I also wanted to ask, should we always use the binomial formula to find p-values when we have a binomial random variable and one population even if the sample size is "large"? Why?

I'm pretty stumped on both of these questions. I know that When the sample size is large, we use the one-sample z-test, and when it is small we typically use the binomial formula to find the p-values.

I would greatly appreciate any help!

"large" is never well-defined. It takes judgment and integrity. It is particularly important when the two results are not necssarily consistent. If it's a result you want (or are being paid for) or don't want, which results will you publish?