These three pillars uphold every implementation of the empirical process control. Without them, Scrum cannot be implemented as intended.

Many people misinterpret the Scrum Guide as stating the Development Team size is limited to 3-9 members.
In reality, it only states that there is inherent risk attached to having less than 3 members and more than 9 members. As the number of members increases, the lines of communication also increase. This can be calculated using the Group intercommunication formula: n(n - 1) / 2 where n is the number of members.
Some teams are able to handle the risk and "synergize" whereas others might struggle. Saying that, the relationship between defining a Sprint Goal and Development Team size is unclear.
But the relationship between the ordering of the Product Backlog, the PO having clear objectives, and the Sprint Goal are direct. The Product Owner typically comes to the Sprint Planning with a business objective in mind and Product Backlog items related to the business objective. After deciding what can be done for the upcoming Sprint, the Scrum Team will craft a Sprint Goal that would be met through the implementation of the items. This is not dependent on the size of the team nor length of the Sprint.
Scrum is also a framework that's fit for purpose. Some projects/products are not fit for Scrum... or, better stated, Scrum is not suitable for all projects/products.