Is it OK to route a trace through a pad?
There is no problem routing traces through pads (like you have done). Be aware when routing power/GND of the current that will be travelling through these traces. This will dictate trace thickness. In addition, search for &quot;power planes&quot;, &quot;ground pours&quot; for more information.
I can see where your confusion may come from. I'm not a fan of how Eagle renders traces connecting or passing through vias/pads.
When you do this:

This is how the copper will actually look on your PCB:

The thickness of the annular ring needs to be considered to ensure that it can carry the required current.