The last few times I sent a TR workout to my Garmin, it didn’t progress through the workout until I pressed the lap button. Is this an expected behavior?
Yes
If the workout has reps (or intervals) within each set, then the intervals will auto-advance. If each set has only one interval, then you will manually advance the laps 
This excerpt from our Help Center explains the difference in depth:
I pulled this from our Help Center article on the topic. Check it out if you’d like to read more.