That only works if you (and any third party code that might run on such a domain) are completely consistent about always specifying the domain as one of your subdomains whenever you set a cookie.

And if your marketing/SEO/business people are ok with having something like "prod" as a subdomain for all your production web pages.

Usually it's mainsite.com for the marketing site, and then app.mainsite.com for actual production, or if you have multiple it'll have the product name, like coolproduct.mainsite.com

We then have app-stg and app-canary subdomains for our test envs which can only be accessed by us (enforced via zero trust). No reason for marketing or SEO teams to care in any case.

When was the last time you saw a public website like that? prod.companyname.com websites are extremely rare especially outside tech.