How to run backward elimination in $R$ with both categorical data and numeric data? Usually in backward elimination, we start with the full model of all covariates, check the $p$-value of $t$-statistic for each covariate (which is compared between the full model and the model minus the given covariate), and remove the covairate with the highest $p$-value until the maximum of $p$-value is all below some threshold $\alpha$.
What about when the data has both categorical (of multiple levels) and numeric data? I know the most natural thing is to do a similar procedure, but now with the $t$-statistic replaced with the $F$-statistic, and compare the entire group of covariates against the full model to eliminate (we remove the categorical variable of all levels all at once). But how can I easily implement this in $R$ and if there is already some package that integrates the scenario. In the usual case of all numeric covariates, I can perform something like
run_backward_elimination = function(alpha){
    S = 1:p
    while(TRUE){
    pvalues = summary(lm(highway_mpg~[,S]))$coefficients[-1,4]
    if(max(pvalues) <= alpha){
      break
    }
        remove_ind = S[which.max(pvalues)]
        S = setdiff(S,remove_ind)
    }
    XS = X[,S,drop=FALSE]; colnames(XS) = S
    print(summary(lm(Y~XS))$coefficients)
}

Then it should be fine, as I can directly check the $t$-static at the summary table of linear regression. But with the Anova table, the $F$-statistic is compared among the inner models, not against the full model like $t$-static does. So I have to manually  run run $p$ many regressions to remove one covariate, which has a rather high complexity and is also tedious to implement.

I would argue that the variable under consideration is the entire factor with all levels; therefore the appropriate test is a "chunk" test of the entire factor variable, not individual levels. In a linear regression, the standard chunk test is the F-test of nested models...
...but stepwise procedures like this are known to be problematic, which just come up on Meta. If fact, the most downvoted post I have seen is a self-answer that argues in favor of stepwise procedures after many members argued against stepwise procedures.
While this is on a Stata website, the discussion is purely regression theory, and I think Harrell makes a compelling case against stepwise procedures.
