Security permission to only read SSAS Cube with Excel
In order for a specific role to have permissions over a cube data you have to specifically grant cube permissions to that role.
That can be done using the Cube tab in the role properties page: in SSAS

-> Databases -> Your db -> Roles -> your specific role -> Cubes tab.

You can assign specific permissions for:


Access - None, Read, or ReadWrite
LocalCube/DrillthroughAccess - None, Drillthrough/Drillthrough and Local Cube
Process


Same for dimensions and their data and specific cell data.
You can find further info about cube permissions in the following articles:


Permissions and Access Rights (Analysis Services - Multidimensional Data)
Configuring permissions for SQL Server Analysis Services
Security with Analysis Services - Overview