It may be helpful here to distinguish three different statistics. The population standard deviation $\sigma$ is given by:
To calculate that we need to know the values of $x_i$ for the whole population. The standard deviation of a sample can be calculated using exactly the same formula, albeit with the $x_i$ being the sample data, $\mu$ the mean of the sample and $n$ the sample size. No "restriction" is needed here. Comparing the calculation of the population and sample standard deviations, therefore, the question why there is no "restriction" in the population formula does not arise.
Where the denominator is required to be $n-1$ rather than $n$ is in the common situation where we use sample data, not to find the standard deviation of the sample, but to estimate the standard deviation of the population. Thus the formula becomes (assuming the population mean is not known so must also be estimated from the sample data):
What can possibly cause confusion here is that this formula is sometimes referred to as giving the sample standard deviation, a phrase which - to me at least - suggests the standard deviation of the sample, not an estimate of the standard deviation of the population.
Thus formula (1) is not the population equivalent of (2), and there is no reason why (1) should have $n-1$ in the denominator just because (2) has.