I start by communicating two very simple premises to the players while they are thinking up their characters:
1. Your characters all know each other, have history together that we will flesh out, and trust each other at least enough to go on dangerous adventures together. They don't have to be best friends, but they've got each other's backs. Establish your character ties accordingly.
2. Regardless of alignment, class, background, backstory, race, or personal characteristics, your character has a reason to engage with the content that I am presenting. You are free to establish what reasons those are.
Then, I provide a list of "character ties" taken from Dungeon World for the players to establish with the other PCs. In some campaigns, playing up those character ties is worth Inspiration. Alternatively, if the PCs are starting at a higher level than 1st level, I ask them to choose two other PCs and briefly describe an adventure they undertook together in the past.
If you do this, everyone's goals will be in line with the adventure scenario or campaign you're presenting and the characters will have reasons to work with each other. A side benefit is that it also creates a lot of context that the players can use to make decisions for their characters and interact with the other PCs more smoothly.
Personally, I like your new backstory system better than the character ties. It makes the characters feel much more tightly knit. However, both accomplish the same goal to different degrees.