Your assertion from the GPL is incorrect
"Essentially, under section 2 of the Terms and Conditions of Copying, Distribution and Modification, if you happen to utilise a piece of code that has been distributed under the GPL - just one piece - then the whole of the final released code that includes that GPL code in it also has to be distributed under the GPL."
falls foul of the most common misconception of the GPL. You do NOT have to license all of your code under the GPL if you use a chunk of GPL'd code. If you read the GPL carefully, you will see that:
"if identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works."
This means that if your program uses a chunk of GPL'd code and your program is not a derivative work, then you are not restricted at all - you can license it in any way you like. You only need to acknowledge and distribute the chunk of GPL'd source code you used.
However, the GPL goes on to clarify:
"But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it."
This is saying that if you create a program which is BASED ON a GPL'd program, then you must distribute the whole program under the GPL. This is quite diferent from just using a chunk. For example, if you create an enhanced word processor program which is based on an existing GPL'd word processor program, then this is regarded as a derivative work and must be distributed under the GPL.