There is not any OOTB feature to automatically add caption to your video, you need to add a parameter in DM component.
Append the copied URL of the video with the following syntax so you can associate it with the copied URL to your caption file:
&caption=<full Copy URL path to the caption file .vtt>,1
Note the ,1 at the end of the caption URL path. Immediately following the VTT filename extension in the path, you can optionally enable or disable the closed caption button on the video player bar by setting to 1 or 0, respectively.
You need to validate the exactly the same path on publish environment & vtt option on publish environment. If it has the correct path then I don't see a reason to not work on publish. Because it does not depends on author/publish it is all together a diff domain from where video is rendered. Additionally, you should also check for console errors.
