AWS - Can a Private Subnet reach the Internet only with NAT without Internet Gateway

Solution 1:

The NAT Gateway needs Internet Gateway to reach the internet.

Technically the hosts in the private subnet don't need IGW to reach out but the NAT GW does, so ... you can't really do it without IGW. Unless you do some creative routing with multiple VPCs, some with and some without NAT/IGW. Perhaps using Transit Gateway? But that's a whole new level of complexity.