Is it Important to Understand Each Normal Form?
Codd formulated the normal forms as a tool to help evaluate your data model for correctness:


  "To free the collection of relations from undesirable insertion,
  update and deletion dependencies; To reduce the need for restructuring
  the collection of relations, as new types of data are introduced, and
  thus increase the life span of application programs; To make the
  relational model more informative to users; To make the collection of
  relations neutral to the query statistics, where these statistics are
  liable to change as time goes by."


You are absolutely correct that if you design your data model and the resulting schema very well to begin with, it will already be in at least 3rd normal form.

That said, I believe that it is crucial to understand the relational model, and the normalization rules as helping guides. When you say:


  "I can accomplish this easily by arranging the data so that the
  columns in each table, other than the primary key, are dependent only
  on the whole primary key."


You are using the 3rd normal form. 
A relation in 3NF, must also be in 2NF and 1NF.

You are asking if you need tools to validate your design, which is something no one can answer. 
If you design a car, will you use available tools to validate that it is safe, fast, and reliable? or will you just trust that your original design achieved these goals without testing it with any tools?