Hello

Can a public key be generated without knowing the private key? yes

https://doc.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/ell_generic.html#

Can the coordinate of a desired number in the curve be generated pubkey at the same time? yes.

After creating the curve, it is possible to create it with the "lift_x" function.
If the source you mentioned has private keys
and if the diagram you show is produced, we can talk about a security vulnerability.
However, it is possible to calculate the points that will form this diagram without private keys.

Thank you.