Trauma induced overreaction?
Eating and swallowing a metal ring (or even one of these balls in the flashback) is a very low risk event, unless there are very sharp edges. The risk is basically of the object remaining stuck in the esophagus, most common in children, very low probability in young people (and only with bigger rings). Inducing vomit, being a chaotic event, not only have a few chance of dislodging the ring, it have the chance to irritate the esophagus.
The other chance is choking if the object end and remain stuck in the other way. In this case full objects like the balls are way more risky then flat and hollow objects like rings where there can in most case still be a way for air to pass, albeit there can be various discomfort and irritations. The flashback implicate Nagi was choking on one of the balls on the ground (albeit gasping and coughing can be seen also in non choking airways blockage (like laryngospasms), albeit I don't undesrtand if cought is a direct symptom or that you can cought during it but it's not a direct cause (some sources seems to imply the second), while gasping without coughing can be an agonal breathing seen in cardiac arrests, but doesn't seems to be the case at all).
Inducing vomit in a choking scenario is mostly useless (the exceptionally small possibility that it can help dislodging the stuck object isn't worth the wasted time for starting more useful manoouvers, considering that in full choking scenarios time is a strict constraints) and can be hurtful (people can choke or "breathe" their own vomit, a medical emergency that can destroy your lungs). This is why vomit is induced in specific positions to secure airways, and not with the face up.
So yeah this scene was an action with 0 usefulness (even if she was choking), risky, and any intervention was totally unnecessary (becouse she wasn't choking). Even if it was stuck in the esophagus, it may warrant manual removal, but it's not a medical emergency, so there is no need to act on your own (or calling an ambulance) as there is no hurry to act. Go to an ER/Hospital on your own.
So yeah overreaction (unless the author is convinced that this is a proper way to help someone that is choking, and in this case I propose we gift them a full basic life support course).
Some sources define choking as a full blockage, others seems to allow partial blockages, and while I'm not sure this is completly correct I'm using this last definition for brevity.