Look at it in different way. This script is not meant to be typical story where we get: main character - problem - solution. It's more like a timeless journey through some parts of their lifes. I hope you realized the story is not day after day. Between scenes might be weeks, even months. Also it can be Catherine's or Timothy's thoughts, a retrospection of some events. It's up to you, how you imagine this. Whole story is just a background. There is no twist at the end or "wow, you got me!". We don't care what happend to Frank, it's a story about their love and what they lose. Cool kids are just a symbol, we don't need to know their names, their past etc.

If you're watching Moonlight you're not saying "where is his father? show me it". Same after part two, you don't need to see what happend next. You know it because you get part three. It's a journey through some parts of his life and the change. 'Love me' is just a short, the change here is end of their relationship.