iTunes library .xml vs. .itl
It sounds like the reason you've run into problems is that you've tried to manually migrate across your iTunes data from your Time Machine Backup.

Without knowing exactly what you did, if you copied the entire iTunes folder from your TM backup to your new Mac, one thing you can try is to tell iTunes where that folder was copied to. To do this you would:


Open iTunes
Go to iTunes > Preferences (or just press command,)
Select the Advanced tab
Now you'll see an option where you can set your iTunes Media Folder location


Migration Assistant

Since you're migrating to a new Mac, you'd probably be better off starting again and using Migration Assistant to transfer all your data across. This will give you the option of migrating what you choose from a Time Machine backup or directly from your other Mac (if you have a way of connecting to it). Migration Assistant also provides other options.

To use Migration Assistant:


On your new Mac, open Migration Assistant (by default it's in your Applications > Utilities folder)
If you happen to have another user logged in, you'll be prompted to log them out first. If not, you'll see an Introduction screen instead.
Click on Continue
You may be prompted to enter your password - if so enter it and click OK
You'll now be asked how you'd like to transfer your data.
Select the option applicable to you (most likely the first one which allows you to transfer from a Time Machine backup)
Click Continue
Follow the prompts to migrate the data you want


Note: This process may take some time.

For more information see Move your content to a new Mac.