Let's skip calling it volatility and variance. Let us deal with variance and standard deviation.
For normally distributed variables, it is very important to distinguish between the true variance and the estimator of the variance and the estimator of the standard deviation.
Variance in its raw form is important in defining the statistical law that governs the sample data, while the standard deviation plays no role (unless you pretend to square it). Variance is the second central moment of the normal distribution. Variance, in its raw form, defines all mean-variance models.
That is great if you know it, but if you do not know it, then the estimators are a bit more of a headache than most people realize.
Finance models such as the CAPM, APT, Fama-French or Black-Scholes are strictly Frequentist models. Constructed in a Bayesian space, the results won't come out the same at all. So we need to remain inside Frequentist conceptualizations of a mean, a variance, and a standard deviation.
The minimum variance unbiased estimator of the mean is $$\bar{x}=\frac{\sum_{i=1}^n}{n}.$$ This is also the maximum likelihood estimator.
The maximum likelihood estimator of the variance is $$s^2_{MLE}=\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n}.$$ Jointly, with the sample mean, this is the most efficient estimator of the variance, but it is a biased estimator. As we are in the Frequentist mindset and not Fisher's likelihood-based model, we have to have an unbiased estimator. Because it would take up a lot of space, I am not going to provide a proof, though it is easy to find online, but the unbiased estimator of the variance is $$s^2_{MVUE}=\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}.$$
So far, so good, we are in the same place as a first-semester statistics textbook. But what is different is that the unbiased estimator of the standard deviation is not $$s=\sqrt{\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}}.$$ Instead, this is the sample standard deviation. It is a biased estimator because the square root is a concave function and it distorts the estimator. It isn't generally true that $$\mathbb{E}(f(\tilde{x}))=f(\mathbb{E}(\tilde{x}))$$.
There is a somewhat complicated correction required to make it unbiased. For small samples, the difference is quite large, but for very large samples the impact is nominal and could be ignored.
The issue is somewhat technical, but these models are somewhat fragile and depend strongly on how they are constructed. The mean-variance models are not supported empirically. They are built inside an axiom system and really do depend on it.
It might be better to think of volatility as the scale measurement of a distribution. For the normal distribution, this is the variance. However, not all distributions have a variance, but many still have a scale parameter. The variance is a somewhat narrow and very important conceptualization of scale. Where it exists it has deep mathematical meaning.
Informally, using the sample standard deviation is usually harmless, but if you are really modeling large sums of money, the small effects turn into real amounts of money. It is technically wrong and on large sums it is also a lot of money.
How in the world does variance take into account the IV of all strikes? If I take the IV of a 30d Call and square it, suddenly that gives me more information than just the IV itself?
If the theory is literally true, then the variance fully describes both all errors of commission and all errors of omission in every state of nature. This is why people like working with the normal model. You do not have to do anything else. Because the square root is concave, you actually do lose information within the theory. The statistic wouldn't be sufficient for the parameter. A statistic is sufficient if $$\Pr(\bf{X}|\boldsymbol{\theta})=\Pr(t(\mathbf{X})|\boldsymbol{\theta}),$$ where $\bf{X}$ is the sample, $t$ is some function of the data (a statistic), and $\boldsymbol{\theta}$ are the parameters. This isn't going to be true, so the information in the sample about the parameter is not the same as information in the statistic about the parameter. It can only be less.
