Yes an injury can help you raise awareness, it can be a tool for learning.
But no, injuries are not really a big enough “part of training” that they should be considered a “part of training.”
If you, your coach, your trainer, your physiotherapist or anyone helping guide you in your physical movement journey is frequently injured, now may be time to inquire as to why?
If the questions aren’t being answered, it may be time to do some further research.