Gateways

Even though the name gateway makes it look special, it's just another IP (need not be .1) of the subnet and this IP is given to a router so that it can then connect this subnet to another networks.

mental model for gateways

The data link layer of a subnet is extended by making a router part of the network.

This is exactly why any broadcast messages reach the gateway interface as well.

Software behind Gateway

Whenever a gateway is created, there is a software behind it that handles and routes all request coming in and going out.