Current answer is adopted as the best

Method 1: Use third-party applications
Step 1: Download and install third-party applications
Search the App Store and download a file management app like Documents by Readdle. The app has a built-in browser that makes it easy for us to download and manage files.

Step 2: Visit the download site using the built-in browser
Open Documents by Readdle, click the browser icon in the lower right corner, and visit a website that supports YouTube video downloads, such as y2mate.com or savefrom.net.

Step 3: Get the video link and paste it
Open the YouTube app, find the video you want to download, click the share button, and then choose to copy the link. Go back to your Documents by Readdle browser and paste the link into the input box on the download site.

Step 4: Select Format and Download
Select the format and quality of the video you wish to download and click the download button. Once the download is complete, the video will be saved in the Downloads folder in Documents by Readdle.

Step 5: Move the video to an album
In Documents by Readdle, find the downloaded video, tap the three dots in the lower right corner, select Move, and then select the folder in the Photos app. This will allow you to find and play the downloaded video in the album.

Method 2: Use shortcut commands
Step 1: Add a shortcut command
First, make sure you have the Shortcuts app installed on your iPhone. Then, visit the iCloud link and add a shortcut command called "Download YouTube Video".

Step 2: Get the video link and run the shortcut command
Open the YouTube app, find the video you want to download, click the Share button, and select Copy Link. Open the Quick Commands app and run the "Download YouTube Video" quick command.

Step 3: Select Format and Download
The Quick Command automatically processes the link and provides download options. Select the video format and quality you wish to download and wait for the download to complete. The video will automatically be saved to an album.

Method 3: Use an online conversion tool
Step 1: Access the online conversion tool
Open Safari browser and visit an online tool that supports YouTube video conversion and download, such as onlinevideoconverter.com.

Step 2: Get the video link and paste it
Open the YouTube app, find the video you want to download, click the Share button, and select Copy Link. Go back to your Safari browser and paste the link into the input box of the online tool.

Step 3: Select Format and Download
Choose the video format you wish to convert and download, then click the Start button. After the conversion is done, click the Download button to save the video to your device.

Step 4: Save the video to an album
Once the download is complete, click the download icon in your Safari browser, locate the downloaded video file, select Share, and then select Save Video. The video will be saved to an album.