but I believe we have more points in favor of its implementation than its omission. I'd love to hear your thoughts. :)

I don't think that we should implement the feature. Indeed, there is not a single variable/feature importance that rules them all and I think that we should not have a single attribute.

Right now, our API of the SelectFromModel is a bit limited: we accept a callable. However, it takes only a fitted estimator as an attribute. So you could still do something ugly as:

from sklearn.feature_selection import SelectFromModel
from sklearn.ensemble import HistGradientBoostingClassifier

X = [[ 0.87, -1.34,  0.31 ],
     [-2.79, -0.02, -0.85 ],
     [-1.34, -0.48, -2.55 ],
     [ 1.92,  1.48,  0.65 ]]
y = [0, 1, 0, 1]

X_test = X.copy()
y_test = y.copy()

def importance_getter(estimator):
    from sklearn.inspection import permutation_importance
    results = permutation_importance(estimator, X_test, y_test)
    return results.importances_mean

selector = SelectFromModel(
    estimator=HistGradientBoostingClassifier(),
    importance_getter=importance_getter,
).fit(X, y)

The broader issue here is that we need an API discussion on how to compute variable importance as discussed here: #20059 (comment)

I think it could useful to find a way to compute the mean decrease in impurity (MDI) but the feature_importances_ API is too limiting because it forces us to only consider training set statistics.

So let's find a way to implement/fix #20059 in general (and in particular how to compute MDI on test data #20059 (comment)) and then we can discuss how to make this available for HistGradientBoosting* models.