How to set up SPFx customizations on a shared tenant?

... for updating apps, I need to remove apps and re-add them back again, per site which is really painful.

I don't think, this is the case. Every time you add a new app package version to the global App catalog, all sites will get the latest updates automatically.

I will assume you are talking about the App customizer customizations. Headers, footers, custom actions, styles, etc.

Option #1: Deploy App globally

• Create tenant-wide SPFx solution
• In your SPFx code, in JavaScript/TypeScript, check the current site's naming convention or site's property bag value or something else that can tell you to which organization the site belongs.
• If condition is not met, do nothing.

Option #2: Require App installation per site

• Require app installation for every site
• As part of the site provisioning process, make sure that the app your custom app is installed

For both options #1 and #2, you don't need to worry about upgrading every single app instance every time you upload a new version to the App catalog. This should be handled automatically. JavaScript bundle is stored in a single place anyway, inside the App Catalog site.