Mac shutting down with 30% battery

You current full charge capacity is 2852 mAH. Full new battery capacity is listed as 4790 mAH. 2852 is 59.5% of the full design capacity. That's how the 59.5% you see is calculated.

Apple recommends you replace the battery when your remaining capacity drops to 80%. You're well below the 80% threshold for replacement.

It says "Mac OS battery status: Check battery"

That's coming from somewhere.

If your Apple menu -> System Preferences/Settings Notifications has certain notifications turned off, you won't see battery status automatically. Generally if you click on the battery icon in the menubar, it will tell you Check battery as well or Service Recommended. If it doesn't, then Cocoanut battery is detecting some precondition that doesn't alert the OS right away.

There is your reason. The battery is over 4 years old. Time to replace. Most batteries last 2-5 years depending on performance and usage.