There is no indication at all in your screenshot, that the photo in question is a Live Photo. Only Live Photos have the controls you are looking for.
Has the photo in question been imported as a Live Photo or only as a still frame? Does it show in the Live Photo album? As a test, use the command "File > Export > Export unmodified original" for one of the Live Photos with the missing controls. This command should at least export two files - the high resolution still frame and the video component. Are both files there? Can your Mac play the video component?
Where is your Photos Library stored? Is it on an external drive or on your startup volume?
Are you using iCloud Photos to sync the Photos Libraries on your iPhone and the Mac ? Or My Photo Stream?
If you are using My Photo Stream, your Live Photos will be transferred as still frames and not Live.