How can I rent or buy a show without a monthly subscription to Apple TV+?

Rent or buy movies and TV shows are separate from the subscription content. One does not require the other. They also have different catalogs: Apple TV+ subscription shows are generally not available for buy or rent, with a few recent exceptions (e.g. Killers of the Flower Moon, Napoleon). Reversely, buy or rent content is generally not included in the Apple TV+ lineup.

Buy movies and TV shows from the Apple TV app - Apple Support

Rent movies from the Apple TV app - Apple Support

So, if it's an AppleTV+ show then I can't purchase/rent that show all on its own?

There is very little overlap. For most of the Apple TV+ catalog, Apple has exclusive distribution rights for its subscription service. For titles from that are available through other resellers as well, then you could buy or rent those through Apple (iTunes Store).

Only recently has Apple begun offering select new Apple-published movie titles as buy/rent (second, after a theatrical release), before bringing said title to the subscription service.

About Apple TV+ - Apple

Apple Originals - Apple TV+ Press

List of Apple TV+ original programming (episodic/series) - Wikipedia

List of Apple TV+ original films - Wikipedia

See the Apple TV+ tab in the Apple TV app.

Buy/rent options are available for other titles from other studios.

See the Store tab in the Apple TV app, if available.

If your device has the Apple TV+ app (note the Plus in the app name), instead of the Apple TV app, then that app is specialized to (limited to) subscription content, and that app wouldn’t feature a Store tab (nor Library). This edition difference is not user choice, but how the device manufacturer and Apple agreed upon.

Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support

just by using the appletv app while signed in with an appleID with a valid means of payment

Apple TV app - Apple

though some smarttv seem to have appletv app which only include appletv+ (subscription) and not rental/purchase of normal itunes store items