Webpack doesn't include js file in dev development - why?

Question

webpack

Set up a build with webpack-server:

webpack.config.js

const path = require('path')
const HtmlWebpackPlugin = require('html-webpack-plugin')

module.exports = {
  entry: './src/main.js',
  output: {
    filename: 'app.js',
    path: path.resolve(__dirname, './public'),
    publicPath: '/public/'
  },
  module: {
    rules: [
      {
        test: /\.js$/,
        loader: 'babel-loader',
        exclude: '/node_modules/'
      }
    ]
  },
  plugins: [
    new HtmlWebpackPlugin({
      hash: true,
      template: './src/index.html',
      filename: './index.html'
    })
  ],
  devServer: {
    contentBase: './src',
    overlay: true,
    historyApiFallback: true
  }
}

When running the webpack-server, the js file is not embedded in index.html, but when the production build is built, everything is embedded as it should.
What didn't I learn?
I’m setting it up not for the first time and it always worked, but here I’m breaking my head for an hour.