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where $\hat\sigma^2$ is the unbaised estimator and not the MLE. See summary.lm

This is the inverse observed Fisher information for $(\beta_0, \beta_1)$ conditional on $\hat\sigma^2$. Now the inverse observed Fisher information you compute is for the triplet $(\beta_0, \beta_1, \sigma)$. I.e., you use the MLE of $\sigma$ and not the unbaised estimator. Thus, I gather the standard errors should differ by factor $\sqrt{n/(n-3 + 1)}$ or something similar. This is the case

where $\hat\sigma^2$ is the unbiased estimator and not the MLE. See summary.lm

This is the inverse observed Fisher information for $(\beta_0, \beta_1)$ conditional on $\hat\sigma^2$. Now the inverse observed Fisher information you compute is for the triplet $(\beta_0, \beta_1, \sigma)$. I.e., you use the MLE of $\sigma$ and not the unbiased estimator. Thus, I gather the standard errors should differ by factor $\sqrt{n/(n-3 + 1)}$ or something similar. This is the case

So to answer

I understand from my readings on the web that optimization is not a simple task but I was wondering if it would be possible to reproduce in a simple way the standard error estimates from glm while using optim.

then you need to scale up the standard errors in the Gaussian example you use.

summary.lm
#R function (object, correlation = FALSE, symbolic.cor = FALSE, 
#R     ...) 
#R {
#R    z <- object
#R    p <- z$rank
#R    rdf <- z$df.residual 
#R    ... 
#R    r <- z$residuals
#R    f <- z$fitted.values
#R    w <- z$weights
#R    if (is.null(w)) {
#R         mss <- if (attr(z$terms, "intercept")) 
#R             sum((f - mean(f))^2)
#R         else sum(f^2)
#R         rss <- sum(r^2)
#R    }
#R    ...
#R    resvar <- rss/rdf
#R    ...
#R    R <- chol2inv(Qr$qr[p1, p1, drop = FALSE])
#R    se <- sqrt(diag(R) * resvar)
#R    ...

obj <- qr(X)
sqrt(sum(resid(m)^2)/n * diag(chol2inv(obj$qr)))
#R [1] 5.71058285 0.09732149

summary.lm
#R function (object, correlation = FALSE, symbolic.cor = FALSE, 
#R     ...) 
#R {
#R    z <- object
#R    p <- z$rank
#R    rdf <- z$df.residual
#R    ...
#R    Qr <- qr.lm(object) 
#R    ... 
#R    r <- z$residuals
#R    f <- z$fitted.values
#R    w <- z$weights
#R    if (is.null(w)) {
#R         mss <- if (attr(z$terms, "intercept")) 
#R             sum((f - mean(f))^2)
#R         else sum(f^2)
#R         rss <- sum(r^2)
#R    }
#R    ...
#R    resvar <- rss/rdf
#R    ...
#R    R <- chol2inv(Qr$qr[p1, p1, drop = FALSE])
#R    se <- sqrt(diag(R) * resvar)
#R    ...

obj <- qr(X)
sqrt(sum(resid(m)^2)/(n-k+1) * diag(chol2inv(obj$qr)))
#R [1] 8.0759837 0.1376334