For Firefox I have my own little extension that just loads a HTML page with more or less the same as what you're describing. I presume you can also do that with Chrome, but I don't use it so idk.

I don't really get why this isn't just a setting, but ah well. There are some extensions that do this out there, but I found all behaved wonky. I'm not sure if they're "doing it wrong" or because I have it easier as I don't need any settings and can just load a static HTML file. I'm not an expert on any of this.

For those interested, in manifest.json I have:

  {
      "manifest_version":          2,
      "name":                      "newtab",
      "description":               "newtab",
      "version":                   "0.5",
      "browser_specific_settings": {"gecko": {"id": "XXX@XXX.XXX"}},

      "chrome_url_overrides": {"newtab": "newtab.html"},
      "permissions":          ["clipboardRead"]
  }