Why do I need a Nat Gateway for ECS containers using AWSVPC?

So here is a scenario:

I have a route table that routes all traffic to the internet gateway, and I have a subnet associated with that route table, which makes it a public subnet. Now I create another route table and I route all its traffic to a Nat gateway that I created I created in the public subnet abovesubnet. Now I create another subnet in this private route table, and associate that with my ECS containers. So in effect, this is how the traffic is being routed:

But what is wrong with doing this:

Why doesn’t associating it with the internet gateway directly work? The first one seems like the same thing but with extra steps. I would really like to know why exactly that is necessary, and what happens "under the hood" in both cases, and what causes the second way to not work.