How to pre-populate dynamic filter on Dashboard tab

I understand that you want to have a set of foundational reports and don't want to replicate them N times for each different Product Team.

I have this exact situation in place with one of my clients where for some Roles, staff is typically associated with a branch, and they are typically only interested in records for their own branch. 

I suggest that you create a table a users whee the Key Field is either the users email address or the userid.

Then you load that table with the employees preferred Product Team(s).

then you make a relationship to any of the details tables which end up being displayed as reports on the dashboard where the reference field on the relationship is a formula for the the current user

User()

Then you go back to all of your reports and you add a filter that essentially says the records need to belong to that product team or else the current user has no preferred Product Team loaded. That will cover the situation for higher level managers who cross all Product Teams. 

So that your challenge is to determine how you maintain this table of users where the key field is the user ID or the users email address and keep it updated as new employees get on boarded and belong to different product teams.

If you don't have a source available to keep that employee list up-to-date, there are strategies to be able to create a button on the dashboard where when a user presses that button they would add them self to that table and be able to adjust their preferred product teams.