Just a quibble, but you say that you "can calculate the annual volatility by taking the standard deviation of the daily returns and scaling it by $\sqrt{252}$, and this should equal exactly 10%." I think it should read that the "annualized standard deviation will tend towards 10%".
The "tends" is actually important, thus "central tendency". The central limit theorem (CLT) states that as sample size $N$ increases from a random sampling of i.i.d. variables, with sample mean ($\mu_s$) and variance ($\sigma^2_s$), that sampling will converge with the population mean ($\mu_p$) and variance ($\sigma^2_p$). This idea is classically demonstrated through the Lindeberg–Lévy CLT:
$\sqrt{n}((\frac{1}{n}\sum_{i=1}^nX_i)-\mu)\to $ distribution $N(0,\sigma^2)$
where:
$\sqrt{n}$ is sample size
$X_i = \{X_1,X_2...\}$ is a sequence of i.i.d. variables with with an expected value equal to $\mu_p$ and expected variance equal $\sigma_p^2$.
$N(0,\sigma^2)$ is the cumulative distribution function
So, actually, your daily sampling will give you the smoother estimate of your actual variation and will converge to "$\mu_p=5\%$ and annual $\sigma_p=10\%$" quicker than your weekly sampling. In either case, the expected values of sample mean and variance are the same; weekly sampling should not result in smoothing if your underlying distribution is, indeed, normal. Note: the "root time" rule works because the variance scales linearly with respect to time.
If, however, the underlying stochastic process is mean-reverting -- as may exist in stock prices -- your intuition about smoothing of longer time periods actually will bear out (i.e., annualized daily variance will over-state actual annual variance).
