Should I get back with my ex after one year?
I see some serious issues with your description.
‘Her friends have told me she starts dating guys and breaks up with them because "they're just not him"'
This line is very telling in several ways. First off, it says that you're communicating with her friends, and not her. This is always going to be problematic. A big part of those stupid reasons you broke up is very much likely because of poor communication. And yet this is still featured - you're still not talking with each other in a way that advances clarity and function in the relationship.
The second problematic issue is that she doesn't seem to be learning from what she's doing. She's stuck in this cycle of using men to prop up her self esteem, hoping to return to some mythical past that never really existed. After all, if the past with you was so perfect, the two of you would be together.
You say she's really depressed since the break up. That is natural and normal. If she could move on without a care in the world, that would mean she has some sort of personality disorder such as being a psychopath.
That said, think long and hard before volunteering to be in a relationship with someone with serious psychological issues. This dysfunction is extremely stressful. It's also damaging over the long term. It will rob the relationship of the happiness you both deserve. People tend to operate on the lowest common denominator. This means you will be servicing this dysfunction in order to maintain some form of normality. An example of this is the relationship where one person has explosive temper, and the partner and their kids tiptoe around them, hoping to never set off their fury. Nobody can relax and be themselves. Everybody caters to the dysfunction in order to not suffer.
Unfortunately love is never enough to sustain a long term relationship. You need communication, and the ability to work together. You need healthy attitudes, and character. You need shared goals and so much more.
You broke up for solid reasons. Rarely does going back work - you already have questions about resuming the relationship. The past is a proven track record of not working, and mal-adapted interactions. Unless there has been serious personal and inter relational work to alter how the two of you interact, it's much more likely that you're just signing up for more of the same. Love isn't enough to make things work out. Relationships operate on much more than feelings.
I would proceed with the greatest of caution.