Currently, the audio and video imported through "Local Media" and "Album" both support self-importing SRT files.
After importing the media, on the playback page, click the subtitle icon in the top right, and in the pop-up panel there is a "Main Subtitle" option, where you can select the SRT file.

The most common reason for this issue is that the SRT file was chosen through a cloud storage app. Many cloud apps on iOS provide only an empty placeholder file in the file picker. You can confirm this by checking the file size.
To fix this, download the SRT file to a local iOS directory first, make sure it has the correct file size and content, and then try importing it again.