|If you have a LAN, you already have a router.|
Not necessarily. All you need to make a LAN is a switch but you cannot share an internet connection with a switch, for that you need a router.
You can make Windows act as a router, that is effectively what Internet Connection Sharing does. However it can be unreliable and unlike a SOHO router, has no built in DHCP which means that everything has to be configured manually.
Giving Windows direct access to the Internet is not a good idea from a security point of view. You will be far better getting a router and using that to share the Internet out to your LAN. A router will be far more reliable and because it has DHCP, a lot easier to configure and because of Network Address Translation, a lot more secure.