EC2 VPC Block Public Access settings should block internet gateway traffic

Before blocking the public access you should read this https://aws.amazon.com/blogs/networking-and-content-delivery/vpc-block-public-access/ explaing how it works and what options you have. In short, if you would just block all public access it would also block access via existing NAT gateway (not what you want). But you can block public ingress or make exceptions for resources that need to be accessible from the internet. And don't start with your production VPC but test first somewhere where you can affort some trial & errors.