MacBook 7,1 internal optical drive not doing anything

The DVD Drive has died.

It doesn't block the CD, but also doesn't try to pull it in or make any noise.

You can insert media before the system has even booted. When you turn a Mac on, it will do a scan of all attached devices (disk drives, CD/DVD drives, USB, etc.) to look for all the potential bootable media. Macs with these optical drives will make that tell-tale noise of the optical heads moving back and forth looking to see if there's media installed. If you're not hearing that - the drive is dead.

It also seems like my Mac doesn't recognize the optical drive at all; it isn't listed anywhere and system info -> burn disks says it couldn't find any drive for it.

This is another sign that it's dead. The optical drive is attached to the SATA bus, so if the system can see it, it's not working.

Could it be the logic board?

Not likely. As long as it gets power, the drive will "make noise." It's possible that the cable/connector to the logic board went bad, but very, very unlikely. Since the drive is mechanical, odds favor that the drive itself is the problem.

So, you need to replace it (or simply remove it). I'm assuming it's a 2010 MacBook Pro since there was no MacBook made in 2010 and the MacBook Air doesn't have an optical drive. iFixit has a guide for replacing the optical drive - it's fairly simple. Personally, I would just remove it and be done. Optical disks are becoming rare and if you really need one, just use USB DVD drive.