No, i don't see Henry being the only reason. Drake played well as a sophomore backing up Yeldon but battled several injuries over the final few years at Bama and early in his rookie year with the Dolphins. There are some guys who just have bad luck but also guys who tend to be prone to injuries, either due physiology/build, running style, or not doing what they need to for injury prevention (nutrition, proper workout regime, stretching, treatment, staying in shape during offseason, etc). I think Drake falls in the physiology/build category.
Some articles and references below......
Bama Injury History:
http://ftw.usatoday.com/2015/12/alabama-kenyan-drake-injuries-suspensions-playoff
Then in Dolphins rookie camp in June:
http://dailydolphin.blog.palmbeachpost.com/2016/06/15/dolphins-rookie-rb-kenyan-drake-suffers-injury-in-wednesday-minicamp-practice/
And again in August camp:
Then later in the year (Nov-Dec) he was on the weekly injury report for a knee. For a guy not getting the lions share of carries, that's a lot of injuries. Drake has a load of potential and brings a unique skillset, but i can't ever see him being a David Johnson type and handle the ball that many times game after game. With Ajayi, he doesn't have to be. He can play a role more like Darren Sproles and compliment what Jay brings to the table.