When/why not to use studentized residuals for regression diagnostics?

Consider a linear regression $$y=X\beta+\varepsilon.$$ Residuals $$e:=y-X\hat\beta$$ are often used as substitutes for the unobserved model errors $$\varepsilon$$ for validating assumptions such as homoskedasticity of $$\varepsilon$$, normality of $$\varepsilon$$ and other.

When the model errors $$\varepsilon$$ are homoskedastic with variance $$\sigma^2_\varepsilon$$, the residuals $$e$$ have unequal variances: $$\text{Var}(e)=\sigma^2_\varepsilon(I-H)$$ where $$I$$ is an identity matrix and $$H:=X(X^\top X)^{-1}X^\top$$ is the hat matrix. (For the same reason, the residuals are also correlated.)

The heteroskedasticity in $$e$$ can be "corrected for"/"undone" by using (internally or externally) studentized residuals $$\tilde{e}_{int}:=\frac{e}{\hat\sigma_{int}\sqrt{1-h_{ii}}}$$ or $$\tilde{e}_{ext}:=\frac{e}{\hat\sigma_{ext}\sqrt{1-h_{ii}}}$$ where $$\hat\sigma_{int}$$ and $$\hat\sigma_{ext}$$ are internal and external estimates of error variance, respectively.

This heteroskedasticity correction seems to come at zero cost. No estimation or approximation errors seems to be introduced this way, aside from uniform (across data points) scaling due to imperfect estimates $$\hat\sigma_{int}$$ and $$\hat\sigma_{ext}$$, and the computational cost is low. This might suggest we should routinely use studentized residuals instead of raw residuals for regression diagnostics, because this way we remedy the problem of heteroskedasticity of $$e$$ without really sacrificing anything.

Question: Are there any reasons for not using $$\tilde{e}_{int}$$ or $$\tilde{e}_{ext}$$ instead of $$e$$ as a substitute for the true unobserved $$\varepsilon$$ whenever doing model diagnostics regarding homoskedasticity, normality and other common assumptions/conditions?