The following works. This is the main logic of the API (v0.0.1): Using the subtitle file's extension as input format.

ffmpeg -f srt unnamed output.vtt

This should work

# Moving forward, we will refer to this resource as talk-1-en
ffmpeg -i https://www.ted.com/talks/subtitles/id/1/lang/en output.vtt

curl -o 1.json https://www.ted.com/talks/subtitles/id/1/lang/en
ffmpeg -i 1.json output.vtt

But the current approach would result in something like the following and fail:

ffmpeg -f json -i 1.json output.vtt

talk-1-en is:

• A JSON file.
• The server specifies its type as Content-Type: application/json; charset=utf-8.
• FFprobe (and FFmpeg) recognize that it is actually of the format tedcaptions, whether the input is from the URL, 1.json, or an extensionless file.

We could simply not specify the format and let ffmpeg guess its type from the input, but some malformed(?) files like sintel-fr will fail to be processed.
If I understand it correctly, sintel-fr has an incorrect magic number. Specifying the filename helps ffmpeg (and VLC) recognize the file type and process it.

Maybe it's not something we should bother with?