I can't do what I need to on my LG Smart TV.
To watch content from Apple TV on your LG smart TV:
- Verify that your LG TV supports the Apple TV app among its smart apps. This depends on the exact model and its year of introduction. Please specify the exact model.
- If unsupported, then you should be able to use an external TV-connected streaming device, such as e.g. Fire TV.
- Your non-Apple smart phone likely shouldn’t be needed. Unless the smart app asks to do a step for activation on your mobile.
- For the Apple TV app on your smart TV, sign in with your Apple ID account. If you don’t have one yet, then you should create one, to make use of the app.
Apple has released two app editions: ① The full Apple TV app on select LG TVs from 2018 or later, and ② the specialized Apple TV+ app (with a Plus in the app name) on LG TVs from 2016 or later that didn’t get the full Apple TV app. The Apple TV+ app doesn’t have a Store or Library for buy or rent movies. Which edition is available to you is not user choice, but manufacturer choice per model as indicated above.
This doesn’t apply to Fire TV, as that has the full Apple TV app for all supported models.
Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support
Watch on these smart TVs and third party streaming devices - Apple Support
How to watch Apple TV+ and more - Apple Support

for example I can go and see the movie Abigail n then I am able to watch it on my Big TV.
For my region, Abigail (2024) is still on pre-order, awaiting VOD release as buy or rent movie from June 19th. It might be earlier in other regions; not sure.
To buy or rent movies from Apple’s Store, you would click the Buy or Rent button on the Details page for that title in the Apple TV app on any of the supported platforms that support financial transactions. Then the price would be billed to the payment method as linked to your Apple ID account. Your Samsung phone wouldn’t help with that, as it wouldn’t support the Apple TV app.
Consider doing the transaction in the Apple TV app on Windows PC, or on Mac, iPhone, iPad, iPod touch, Apple TV box, Roku, PlayStation (4, 5), Xbox (One, One S, X, S), LG TV, Samsung TV, Vizio TV (instead of Fire TV, Android TV, Google TV). I think those are your options.