Does the gateway engine allow you to teleport to the chambers and engines on other planets when you activate them?
Sort of. The structure behind the gateway network is a little more complicated. There is actually a third structure called a gateway portal. This is above the engine in the hierarchy. You can use a chamber to teleport to an engine and an engine to teleport to the portal. Once at the portal, you can teleport to other gateway engines on other planets.
So long story short, you can use the engine to teleport to other planets, you just have to go around it a long way, teleporting a couple of times in the process. If you need to know more, this guide will cover everything about the gateway engine and activating the network.