This is the correct formula for the variance of a binomial(n,p) distribution but i think you're mis-interpreting it.variance = n*p*(1-p)
This is the variance of the number of shots on target if the model is correct, not the variance of your estimate of the conversion rate. You would expect the variability of the number of shots on target to increase with the number of attempts, which is what you've observed.
For the variance of your estimate, i'd think along these lines (not sure it's correct though):
given that the true distribution is ,and that the estimated conversion rate is .
Which is decreasing in n as you expected. Whether or not this can be inverted to create a significance measure with a useful interpretation, i have no idea im afraid.