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 observations are not always independent. For instance, there may be auto-correlation across data points that are close together in time/space.
In those cases, we need to specify a model that accounts for the dependence structure of the observations in a statistically adequate way, which will ensure that the residuals from the model will be IID. For example, we could use ARIMA models.
Another option is to continue with standard OLS and possibly non-independent residuals, but to correct the estimates of standard errors, using appropriate estimators, such as Newey-West standard errors
