Without being in the situation, I'm not going to pass judgment.
However, I will offer an opinion.
My philosophy on spotting is to spot as a means of teaching the right shapes/positions. As far as doing the entire skill for a kid, I try to do that as little as possible. Now, there are certain skills, particular on bars (i.e. front hip circles, mill circles, etc. etc.) where there are so few lead-ups that you pretty much have to spot the entire skill over and over and over.
One of the things that I see amongst gymnastics coaches are few who are truly great teachers. It's one thing to be a "coach." It's entirely another thing to be one who can truly teach the sport or any sport for that matter. In my experience across numerous gyms, few coaches have a great understanding of the proper progressions and lead-ups for teaching skills. Furthermore, few truly have an appreciation for the amount of time that certain progressions take. It might take 500 reps of a single progression for a gymnast to get it right. Suppose you did 10 per day for five days a week. That's roughly two months on a single drill. I've run into coaches who will do a drill for a couple of practices and then expect results. Sorry, but, it doesn't work that way.
In the situation that's being presented, it's clear that the gymnast is fearful of the series. And, who can blame her? She's being asked to flip on 4 inches that's not very forgiving if you happen to slip a foot and really whack it. With that being said, beam is probably one of the safer events actually because you're not inverted as often. Most falls on beam that I've seen usually just result in a nasty bruise.
The best approach for developing a beam series is to use reverse psychology. My philosophy is to graduate kids onto the higher beams. In other words, if the technique is not correct, I do not push them onward. I am notorious for telling kids to get off of the high beam and get down on the line on the floor or on a low beam. And, I keep them there until I see it done properly there. It makes little sense to try and walk before we run. I believe that if you can't pretty much do at least 9-10 out of 10 of whatever on the line, then the low beam, then middle beam, etc. then you have no business on the high beam.
For most kids, this is pretty effective. Of course, there's always an exception. For those kids, I prefer to challenge and negotiate. "How many do you think you can get done today?" "Bet you can't get 5 done by the end of practice." "Try at least five and let me know how many you make...let me at least watch one..." By only watching one attempt, you can reduce the pressure of the situation. At some point, you do have to push a little and that's true of any skill. Once you've done the proper progressions and the technique is in place, then it's time to go for it. If the gymnast shows perfect technique on the progressions and still can't push themselves through it -- unfortunately, then it is time to consider another sport because the requirements are only going to get tougher. But, you can push without the excess stress and anxiety by just being "matter of fact" and non-emotional in saying that something needs to get done. Then, leave it in the hands of the athlete and if they don't get it done, then they have to deal with the consequences.
Anyways, with the situation that's being presented here, my first question is how good is your daughter's series on the line and low beam? Could she go 9 or 10 out of 10 and hit just about perfect? Since I've seen this scenario played out in the many gyms that I've been in, my guess would be "no." My guess is that this whole thing has been rushed and the coach is just trying to push and ultimately the situation is creating much more stress and anxiety than it needs to. Furthermore, if the technique is not correct, then my feeling is that the body has the ability to sense this. When your body sense danger, it's going to want to stop. I mean, if we didn't have such defense mechanisms, then how would we survive?
The first step in the recovery process is for the coach to back off to reduce the stress of the situation. In many cases, it's the stress of the situation that causes the "balking" as opposed to an actual fear. The second step is to let the gymnast know that fear is normal. The third is to build up the gymnast's confidence as means of making them realize and believe that they can work through the "fear." The final step is giving the gymnast a reasonable amount of control and working together to accomplish the skill. It's not complicated - it just takes a lot of patience and time.