Why assume independent errors and not independent observations when fitting a linear model? When we fit a general linear model, we assume that the errors are independent. But why not just assume that the observations are independent? Isn't it equivalent, and (in my view) more intuitive?

The assumption that the errors are independent means there is no relationship between the residuals of our model (the errors) and the response variable (observations).
You are correct that the observations are also assumed to be independent of each other and the errors should also be independent of each other!
