You can use the Alert If as a filter, just like you described, but it must be on the "trigger" view. The trigger view is the "instruction set" for VizAlerts. It does not matter what it looks like, only what fields and data it contains. Your Alert If tells it when to do something, and the Email To, Body, etc fields tell it what to do when the data is visible. You can embed other sheets in your dashboard, or even from other workbooks on Tableau Server, in the body of the emails you send out, so there is no need to put the trigger view in the actual dashboard you want people to see. Does that make sense?
An example would be if you have two sheets in your workbook, one called "Trigger" and the other called "Dashboard Report". Trigger is an ugly text table that just tells VizAlerts what to do, and has all your special fields on it. Dashboard Report is beautiful and what you want recipients of the alert to receive.
Assuming you're using VizAlerts 2.1, you will need, at minimum, three fields on your Trigger sheet:
For Email Action, you can leave it as '1' if you're using the Alert If filter to remove data when you don't want to send an alert.
For Email To, you can edit it to reflect your team alias, e.g. 'firstname.lastname@example.org', or base it on individual email addresses if you've got them in your data connection, e.g. [Salesperson Email].
For Email Body, you will use a "content reference" to embed Dashboard Report into the email you want to send. It can be a text calc as simple as:
Note that this is a text string, and not an actual Tableau function. VizAlerts will read what it says, and interpret it to mean "Go to Tableau Server, look up the 'thisworkbook' workbook, and embed the sheet 'DashboardReport' into the email as a PNG".
So the result will be one or more emails, sent when your Alert If field is "1", sent to 'email@example.com', with nothing in it except an image of Dashboard Report.
Hopefully this helps.