Sorry, I previously gave wrong information. The Captions can either be added as a separate sidecar file to your video, or to burn in SRT, ITT and CE-608 captions they must be added in the Captions dialogue in the share menu (as you are showing).
Well. I guess I am 0 for 2. I wasn't able to get ITT Captions to burn in the first time, but just tried again and it worked. So, I think you are doing it the right way, just not sure why it didn't work for you. Using latest FCPX 10.4.6.
I know its the right way because I have successfully was able to burn in using a totally different job. It just seems strange. I am on 10.4.4 and thats the same version I used when I successfully burnt in captions.
I was just having this very same problem. Tried making a new library and project and that didn't work.
What I did notice was that the Media I was using already had embedded English captions, and I was trying to burn in Spanish ones. I thought maybe FCPX was getting confused about this. So my fix was to bring in a new export of the finished film with out embedded English captions, and then import our Spanish ones. This time the Burn-in worked.
I just found the reason for not burnt the subtitles in some occasions. Basically starting timecode in the project setting should be 00:00:00:00. Other than this, the subtitle will not burn in. Mine was 10:00:00:00 and obviously I couldn’t make this work until I’ve changed it the 00. That’s it.