Please confirm you are using FH v7.0.8 and Change Any Fact Tag plugin v3.4.
I think you misunderstand the purpose of the Change Any Fact Tag plugin.
It is intended to change the GEDCOM Tag of actual Facts in your database.
It has nothing to do with changing the Tools > Fact Types… definitions that specify how those database Facts are displayed.
The TMG (project) Fact Set imports the TMG Master Tag Type List definitions and they override the Standard Fact Set.
So if you want the imported Birth events to use the Standard Fact Set definition you simply delete the TMG (project) definition for Birth.
Use Tools > Fact Types… and at the top select Fact Set: TMG (project) which is crucial.
Select the Birth event, and hold down the Ctrl key to also select other events, then click the Delete key and acknowledge the warning.
Now there is no TMG (project) Fact definition for those deleted definitions, so the actual Facts will use the Standard Fact Set just like your other Projects.
BTW: You may have identified a flaw in the Change Any Fact Tag plugin regarding the Source Tag Set: TMG (project) that I shall investigate.
However, it does not prevent it from changing any database Fact Tags as there is a simple workaround.
FYI: I am currently in discussion with CP regarding Fact import problems for TMG that they have not perfected in my opinion.