Why is a pooled estimate of the population proportion used in a two-sample hypothesis test ($H_0:p_1=p_2$) for proportions, but not used when constructing a confidence interval for the difference between two proportions ($p_1-p_2$)?

The pooled estimate is only valid when the population means are equal, not the proportions

The confidence interval must use the unpooled standard error to guarantee the interval is always centered at ${\hat{p}}_1-{\hat{p}}_2$

The hypothesis test assumes the proportions are equal under the null hypothesis ($H_0$), allowing a single, pooled estimate for the common proportion, while the confidence interval makes no such assumption

The confidence interval requires separate variance estimates because it aims to capture a wide range of possible differences, even large ones, where the pooled estimate is inaccurate