Hey narkurs,
Thanks for using Apple Support Communities! I understand you aren't seeing Birthdays on your new MacBook Pro as expected. I definitely don't want you to miss any important birthdays, and I'm happy to help.
From the details you provided it sounds like you've followed the steps in this article: Show or hide the Birthdays calendar on Mac.
If that is correct follow the basic Mac troubleshooting steps below so we can work on isolating the issue:
1 - Close all apps. If an app will not quit, follow these steps to force it to quit: How to force an app to quit on your Mac.
2 - Restart your Mac: Go to the Apple menu in the top left corner of your screen > Restart.
3 - Check Calendar and see if the Birthdays are appearing. If there is no change, restart in safe mode and test again: Use safe mode to isolate issues with your Mac.
4 - Restart the MacBook Pro as you normally would. If the Birthdays are still not appearing, create a new user and add the account your Contacts are from. This article will show you how: How to test an issue in another user account on your Mac.
Let me know the results, and if you're able to view the birthdays for your contacts on any other Apple devices you have.
Thanks again, and have a great day!