Advancing the tube to the carina (or deeper) and then pulling back is something I have never heard of.
In fact, this method sounds like it would inhibit ventilation of at least one lung, and possibly both if your luck didn't improve.
You push the obstruction into the right mainstem with the tube, and get a gunky plug stuck in the tip of the ETT in the process, then withdraw, ventilate, and blow the plug into the other mainstem...following the path of least resistance? Does this sound plausible to anyone else, or am I thinking too much into it?
In my experience the proper escalation for obstructed airway is:
If FBO is visible on laryngoscopy, remove object with forceps.
If unable to remove with forceps, move to surgical airway (assuming you see obstruction, but are unable to manipulate and remove it.)
In the event that provider saw the ETT pass through the vocal cords, and the "can't ventilate, can't oxygenate" situation persists despite ruling out other causes (DOPE, costal rigidity due to fentanyl administration, etc.) then proceed to surgical airway.
I am not advocating the "slice first, ask questions later" mentality, however I have seen providers with excellent airway management skills get bogged down in an intubation attempt that went nowhere but downhill after the first miss. They just could not seem to get their feet underneath them.
There are very few airways out in the field that refuse to "play by the rules" but when you encounter one, you better be ready to think on your feet, or else you are going to find yourself in an un-recoverable flat spin pretty quick.
All that is to say, I have never heard of advancing the tube then withdrawing as a correction method for FBO or can't ventilate can't oxygenate, and I think your time would be better spent dealing directly with the obstruction of possible, or bypassing it entirely.