I don't think Mr Laserbeam meant to blame people for using sayings he dislikes. What he meant, I believe, is that sometimes there is a phrase that you really dislike. And if it's a popular one and lots of people say it all the time, it gets even more annoying. It's like screeching chalk on a blackboard. Once in a while is bad enough, but imagine you'll have to hear it all day long...
While "it's gonna be allright / will get better" is a comfort at times, in a situation that you are certain won't improve its use is not really appreciated. It always depends on the situation. For example, it would be pretty foolish to say it to someone who's on the verge of dying. It might, however, be helpful with a kid who fell and hurt a knee.
The problem is, imo, that many people use those phrases without thinking. It's just like saying "Gesundheit!" when someone sneezes. It's something you do automatically, because it's what one does. But you don't consider first that it means "good health to you!" and then very deliberately wish that to the sneezing person.
Thoughtless phrases, if recognized as such, can hurt more than insults because they indirectly tell us that the other person doesn't really care about us but reacts only automatically and without conscious thinking.
If you manage to put some feelings behind the words, it's a completely different matter. Then it's really a comfort