"Is there a way to give a specific group of team members access to every board?"
This is easy to do with a Taskfeed Board Sharing Rule. The Sharing Rule can be based on who owns the Board or a Criteria of the Board; e.g., Board Type. Or just for all boards. If these team members are not in a Public Group yet, you will need to create a Public Group that includes these users.
To create a Public Group:
Go to Setup > Manage Users > Public Groups
Select “Users” under “Search for” to add individual users to a new Public Group.
Now Create the Sharing Rule that allows this Public Group access to every board (or any other criteria):
From Setup, type Sharing Settings in the Quick Find bar.
Scroll all the way down to Taskfeed Board Sharing Rules
Select whether you want to share based on Record Owner or based on Criteria.
To select the users you want to share the rule with, select the Public Group where those users belong to.