The general answer is no it is not a good idea to merely ignore, disregard, or go outside the imprint area. It's there for a reason. That is the area the vendor guarantees will reproduce as anticipated.
....They have responded that I need to keep all non-background content within the 3x3 square smack in the middle...
Then that's the requirement for the vendor you've chosen. It doesn't matter if other products on the market may not adhere to that requirement.
There's a marked difference between "mills" that are designed to do small runs of unique items for many customers... and custom shops that are designed to do large runs for one customer.
The short run shops have automated processes and restrict artwork in order to be able to just print everything the same way. That way they don't have to mess with any custom setups.
While you are absolutely correct that your artwork could be reproduced in spite of it not fitting in that 3x3 square. That would most likely take a custom run and many more dollars. The vendor may have the ability to do custom runs. They'll cost much more. But you could certainly ask them if you really want the art to not fall within that 3x3 square.
The reality is, the vendor may only print in that 3x3 area. (or slightly larger - as in 3-1/8 x 3-1/8" - to ensure everything is visible). They may not even have the ability to print 3x4 with their process.
It's similar for T-shirts..... mills will restrict art to the 12x12" center of a shirt so they can print on existing blank shirts. Custom shops allow you to print anything because they print on the material and then form a shirt from that material. Printing, then making the shirt allows for infinite possibilities. However it's a more detailed, lengthy, and costly processes than just throwing a prefab blank shirt on a screen press and hitting "print".
All that being posted you can certainly approve the artwork even though it's not within that 3x3 square.. it just means if the end product is bad, due to faulty registration or a bad trim, you eat the cost, not the printer.