Imagine the workflow of a dashboard as a chain. Each link of that chain is a widget. Links are joined together by interactions. When you swing one end of the chain, the rest follows. When you open a dashboard, you want to ensure the chain is taunt and all of the links are in-line, e.g. the visualizations are all in sync behind that first widget. That first widget is the beginning of your dashboard workflow, and is often in "self provider" mode.
Some widget types are better than other for the first widget in a workflow. Ideally, you want the first widget to have a default selection, which causes an interaction to be enforced and keep that chain taunt. I personally like the resource widget as a starting point, since you can tag filter it as much as you like, and it has the option "autoselect first row", which selects a resource and keep the interaction chain taunt.
The scenario you describe of widget filtering leads me to believe that you haven't set up tag filtering for all widgets on the dashboard. If you want to limit a point of view for a dashboard audience, all widgets in the dashboard need to have tag filtering configured appropriately.
What you describe is a pretty common request and we can usually accomplish the desired result. It takes a little time sometimes, but when you get your tag filtering strategy figured out everything else will fall in to place.
Mark
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I forget what exact version the option was added to the widget, but 5.8+ should have it.