How to develop a Google Apps Script Web App with different stages / different GSuite accounts?

Does this work?

Yes, just make sure the account has edit access to be able to push (but I am sure you know that).

Is there a better approach?

There sort of is (unless you implicitly meant it), to quote from issue #42 of the clasp project GitHub:

New flag:

clasp login --local Saves .clasprc.json locally, but would use global clasp credentials.

clasp login # global for all projects
clasp login --creds creds.json # your own creds
clasp login --local # local project. Useful for multiple Google accounts.
clasp login --creds creds.json --local # own creds, local

So technically you can use multiple accounts, but in the end, it boils down to your login / logout technique.

Regrading discussion on switching between apps script projects with CLASP, I ended up writing a basic utility script for switching between Apps Script projects to push to (no dependencies, but if you want to manage flags in style, check out the popular Yargs package):

const fs = require('fs').promises;
const { promisify } = require("util");
const { spawn } = require("child_process");

const rl = require("readline");

const promiseAns = () => {
    const dummy = rl.createInterface({
        input: process.stdin
    });

    dummy.question[promisify.custom] = function (query) {
        return new Promise((resolve) => this.question( query, resolve));
    };

    return promisify(dummy.question);
};



/**
 * @summary asks to confirm and exits if ok
 * @param {import("readline").Interface} cons 
 * @param {string} init
 */
const checkExit = async (cons, init) =>{ 

    if ( !/exit/.test(init) ) {
        return;
    }

    const question = promiseAns();

    const ans = await question.bind(cons)(`Are you sure (Y|n)?\n`);

    if (/^Y(?:es)?/i.test(ans)) {
        process.exit();
    }
}

/**
 * @summary repeat question until matches
 * @param {import("readline").Interface} cons 
 * @param {string} query 
 * @param {(ans: string) => boolean} condition 
 * @param {(ans: string) => any} onSuccess 
 */
const askUntil = (cons, query, condition, onSuccess) => cons.question(query, async (ans) => {

    await checkExit(cons, ans);

    if (!condition(ans)) {
        return askUntil(cons, query, condition, onSuccess);
    }

    onSuccess(ans);
});

/**
 * @summary makes readline interface
 */
const makeRL = () => {

    const cons = rl.createInterface({
        input: process.stdin,
        output: process.stdout,
    });

    cons.on("line", (ln) => console.log(ln));

    return cons;
};

process.on("unhandledRejection", (err) => {
    console.log(err);
    process.exit();
});

const APP_CONFIG = {

    paths: {

        dotclasp: "./.clasp.json",

        ids: "./ids.txt"

    }

};

(async (configuration) => {

    const cons = makeRL();

    const { paths: { dotclasp, ids } } = configuration;

    const text = await fs.readFile(ids, { encoding: "utf-8" });

    const lines = text.split(/\r?\n/);

    const relevant = lines.filter((line) => /^(\w+)\s+\w+/.test(line));

    const lookup = {};

    relevant.forEach((lbl) => {
        const [label, id] = lbl.split(/\s+/);
        lookup[label] = id;
    });

    const config = require(dotclasp);

    const [label] = Object.entries(lookup).find(([, id]) => config.scriptId === id);

    cons.emit("line", `Currently selected: ${label}`);

    const { argv } = process;

    const push = argv.includes("--push");

    askUntil(cons, `Select project (${Object.keys(lookup).join(" | ")})\n`, (ans) => lookup[ans], async (ans) => {

        config.scriptId = lookup[ans];

        try {
            await fs.writeFile(dotclasp, JSON.stringify(config), { encoding: "utf-8" });
            cons.emit("line", `switched to ${ans}`);
        }
        catch {
            cons.emit("line", `failed to switch to ${ans}`);
        }

        if (!push) {
            process.exit();
        }

        const cp = spawn(`clasp push --force`, {
            stdio: "inherit",
            shell: true
        });

        cp.on("error", ({ message }) => {
            cons.write(message);
            process.exit();
        });

        cp.on("exit", () => {
            cons.emit("line", `pushed to ${ans}`);
            process.exit();
        });

    });

})(APP_CONFIG);