I think you might be using the term "snowflake" loosely... Snowflake, strictly speaking, requires a single Fact and a number of Dimensions, when some of the Dimensions are linked to other Dimensions and not to the Star.
In a Star model, all 4 dimensions should be linked to the Fact. In a Snowflake, Supplier might be linked to Products and SalesPerson might be linked to Customer. Both models are valid. If the original data is organized closer to the latter option, then you need to add some data modelling logic to bring it to the Star format. That's the extra effort that may be necessary or may be unnecessary - depending on your preference.
What you described as a "snowflake" sounds like an Entity-Relationship model that's coming straight from a transnational system. That's not what we usually call "snowflake".