Omitted interaction terms don't cause omitted variable bias maybe?

interactions.R

library(tidyverse)
theme_set(theme_minimal())
library(broom)

one_sim = function(id, n = 100) {
    dataf = tibble(x1  = rnorm(n, 0, 1), 
                   x2  = rnorm(n, 0, 1), 
                   eps = rnorm(n, 0, .1),
                   y = 1 + x1 + 3*x1*x2 + eps)
    coef_df = lm(y ~ x1 + x2, dataf) |> 
        tidy(conf.int = TRUE) |> 
        mutate(id)
    return(coef_df)
}

set.seed(2021-12-24)

df = map_dfr(1:1000, one_sim)

df |> 
    ggplot(aes(estimate)) +
    geom_density() +
    geom_rug() +
    geom_vline(aes(xintercept = value), 
               data = tribble(
                   ~ term, ~ value, 
                   '(Intercept)', 1, 
                   'x1', 1, 
                   'x2', 0),
               color = 'blue') +
    facet_wrap(vars(term), scales = 'free')

df |> 
    ggplot(aes(id, y = estimate, ymin = conf.low, ymax = conf.high)) +
    geom_pointrange() +
    facet_wrap(vars(term), scales = 'free') +
    coord_flip()