how can "my computer" read an external drive ?

All content needs to be in iTunes. If the content is on an external drive you can either reference it or move your media folder to the external drive. what are your current advanced settings in iTunes preferences.

And here are the homesharing instructions Use Home Sharing to share iTunes content with other devices - Apple Support

Have you followed any instructions, have you restarted.

1. Open iTunes on your computer.
2. Sign in with your Apple ID and password. Use the same Apple ID for every computer or device on your Home Sharing network.
3. Go to File > Home Sharing.
4. Turn on Home Sharing.

1. On your Apple TV (4th generation), go to Settings > Accounts > Home Sharing. On earlier models of Apple TV, go to Settings > Computers.
2. Select Turn on Home Sharing.
3. Then enter your Apple ID and password. Use the same Apple ID for every computer or device on your Home Sharing network.

DO you mean nothing changed on the Apple TV or nothing changed in itunes

To store and use your iTunes library on an external hard drive, do the following:

1. Find and purchase an external hard drive that's in your price range and is substantially bigger than your current iTunes library--you'll want a lot of room to grow into before you need to replace it.
2. Connect your new external hard drive to the computer with your iTunes library on it and backup your iTunes library to the external hard drive. How long this takes will depend on the size of your library and the speed of your computer/external hard drive.
3. Quit iTunes.
4. Hold down the Option key on a Mac or the Shift key on Windows and launch iTunes. Hold down that key until a window pops up asking you to Choose iTunes Library.

1. Click Choose Library.
2. Navigate through your computer to find the external hard drive. On the external hard drive, navigate to the location where you backed up your iTunes library.
3. When you find that folder (on a Mac) or a file called iTunes library.itl (on Windows), click Choose on a Mac or OK on Windows.
4. iTunes will load that library and automatically adjust its settings to make that the default iTunes folder while you're using it. Assuming you followed all the steps in the backup process (most importantly consolidating and organizing your library), you'll be able to use your iTunes library on the external hard drive just like it was on your main hard drive.

Sorry nothing is "obvious" on a user support forum. We are trying to assist you with your problem and the these are the steps that we all go through but many may miss a seemingly innocuous step which can result in non-resolution of the issue you are experiencing...

Can you see the movies from the external drive in iTunes on your Mac/PC?

then you need to add the movies to your iTunes library. choose File > Add to Library, locate a file or folder, and click Open.