You're doing it wrong, it's not about getting through gates, it's killing things which get you to the boss. And you can't speed run, it's a set 15 mins timer.
You'll have better luck posting in the #fireteam section of the forum. This section is for Destiny 1 topics, so you probably won't find too many takers here.
Also, just FYI, your tactic in your post is not how to do a speed run of the Forest. Rushing past enemies is detrimental, as your goal is to get to 100% kills for each branch, and each enemy killed gets you 2% minimum. Running past all the enemies to get to the red doors will make things take longer than if you kill everything along the way.