Confidence Intervals errors

After many many confidence intervals constructed the amount the cover the true proportion should be exactly 95%, but when I do this a million times, I get something like 94.65% or maybe 95.6% depending on what the true proportion was. Im confused because this value should be EXTREMELY close to 95%, like 94.99% but its not and I also did a rough modeling of the situation on desmos and it said the chance I got what I got is 0 if it was truly supposed to be 95%. I think this is because the standard deviations are different as a sample from the population but I want to see if anyone knows exactly why. If this is true then doesn't that make the definition of the confidence interval invalid if its not truly 95%?