Localhost. This frees you of all kinds of limitations. The browser becomes a true OS view port with proper security since you control what third party code (if any) is exposed to your user.
What is normally the server instead becomes the actual local application written in any language, though I choose Node and TypeScript for this. You can still have an actual remote web server (optionally plural) though, your app just proxies to them.
Performance dramatically improves because for most of the content and application code network transmission concerns are eliminated. At that point your application code either performs like a native app or your incompetence is fully exposed.
Your website gains the full expressive capabilities of your operating system plus any application running in a shell.
You can still run the site from a vanity domain with your company’s HTTP certificates. Have a subdomain that points to a loopback IP, such as https://local.I-love-my-users.com pointing to 127.0.0.1 and ::1. You will need HTTP certificates bind associate that domain to those IPs and resolves to the same revocation chain as your other HTTPS certificates for your other certificates.
I strongly you recommend you try it before you complain about running your next user facing site from localhost.