Regression Notation - does the distribution of $y_i$ refer to the distribution of $(Y|X)$? The title question is the main focus of what I am trying to understand, but the reasoning behind my asking is more nuanced.
I have recently made transition from learning about basic forms of regression to studying generalized linear models. In doing so, I've realized some gaps in my understanding fundamentals of assumption analysis.
Namely, most learning resources emphasize the assumption that $y_i$ be normally distributed in order to perform a linear regression.
This is contrary to my understanding, which is that $y_i$ is a random variable that simply represents a single observation in the sample. Its distribution should mirror that of the entire population, which we do not know, but estimate with the sample distribution.
Here is my confusion: In regression, we generally do not care about the distribution of our dependent variable itself, we only care that the residuals of our estimates are normally distributed.
Therefore, to make the assertion that $y_i$ must be normally distributed, the only way this would make sense is if $y_i$ = $(Y|X)$. I have not seen explicit clarification of this, however.
Any insight would be much appreciated, thanks so much!

One way to further explore this question is to consider a data generation mechanism such as $$X\sim N(0,1), Y|X \sim N(\beta X,\sigma^2).$$  What does this imply about the marginal distribution of $Y\sim N(?,?).$  To get this, one can use for example $$E(Y)=E(E(Y|X)).$$
As you said, with regression analysis we typically model $Y|X.$
