Hello,
From your description, my understanding is that you have an existing “template team" with a number of ”Document Library“ tabs, and your problem is that when you create new team based on the existing ”template team", the new team doesn't has the custom “Document Library” tabs, right?
If so, I just tested this on my side, based on the test result, the new created team has all the custom tabs like the existing team, but please kindly note that after I create the new team, I will still need to configure the settings for apps and tabs. Please see:
The existing template team:

The new team:

Given this result, about your problem, please make sure you select the check box before Tabs when creating new team from existing template team, for your reference:

Best Regards,
Jennifer