結論

Google Sheets API だけじゃなくて、Google Drive API も使う

const sheets = google.sheets({ version: "v4", auth: oAuth2Client });
const drive = google.drive({ version: "v3", auth: oAuth2Client });

// スプレッドシートを作成する
const createdSheet = await sheets.spreadsheets.create();

// Google Drive APIでスプレッドシートの権限を作成する
await drive.permissions.create({
  fileId: createdSheet.data.spreadsheetId,
  requestBody: {
    type: "anyone",
    role: "writer",
  },
});

// 誰でも編集できるスプレッドシートのURL
console.log(createdSheet.data.spreadsheetUrl);

動くサンプル

credential.jsonとtoken.jsonはあらかじめ作成してある前提です。
トークンの生成がまだの方はこちらのドキュメントを参考に、クレデンシャルのダウンロードとトークンの生成を行ってください。
https://developers.google.com/sheets/api/quickstart/nodejs

import { promises as fs } from "fs";
import { google } from "googleapis";

(async () => {
  const credentials: GCPCredentials = await fs
    .readFile(`${__dirname}/../credentials.json`)
    .then((b) => b.toString())
    .then((s) => JSON.parse(s));

  const { client_id, client_secret, redirect_uris } = credentials.installed;

  const oAuth2Client = new google.auth.OAuth2(
    client_id,
    client_secret,
    redirect_uris[0]
  );

  const token = await fs
    .readFile(`${__dirname}/../token.json`)
    .then((b) => b.toString())
    .then((s) => JSON.parse(s));

  oAuth2Client.setCredentials(token);

  const sheets = google.sheets({ version: "v4", auth: oAuth2Client });
  const drive = google.drive({ version: "v3", auth: oAuth2Client });

  const createdSheet = await sheets.spreadsheets.create();

  await drive.permissions.create({
    fileId: createdSheet.data.spreadsheetId,
    requestBody: {
      type: "anyone",
      role: "writer",
    },
  });

  console.log(createdSheet.data.spreadsheetUrl);
})();

type GCPCredentials = {
  installed: {
    client_id: string;
    project_id: string;
    auth_uri: string;
    token_uri: string;
    auth_provider_x509_cert_url: string;
    client_secret: string;
    redirect_uris: string[];
  };
};