1) I really don't know where you get this idea from observations of the real world: "the most perfect circle possible in a quantized universe would have a pi of 4". Beyond the fact that your observations of the world do not contain "the most perfect circle possible in a quantized universe", if it did contain such a circle, you would only be able to observe and experiment upon it with real world tools. Remember, on the wiki it is claimed that a Zetetic "bases his conclusions on experimentation and observation rather than on an initial theory that is to be proved or disproved." So, let's do that. Let's abandon our "initial theory" from grade school that Pi has ANY fixed value, whether that be 4, or 3.14, or something else. Maybe every circle has a different ratio of perimeter to diameter, who knows? Let's find out! OBSERVE a round object (and please, let's not pretend we don't know perfectly well what that word "round" means in the common usage) and perform an EXPERIMENT upon it, measuring its circumference and diameter. Repeat with another round object. Go again, and again, and again. Do not speak of plancks, you cannot observe them and therefor they are the very definition of an "initial theory." And if there are no plancks, as far as you can tell, then the perimeter of each round object is exactly what you measure it to be. I have done this experiment as covered in a previous post, and I urge anyone still reading along with us to conduct the experiment yourself. No doubt you can find at least three different round objects in easy reach, go measure them.
2) You seem to hold Zeno as some sort of pinnacle of Greek science. Your statement "the Ancient Greeks...whose science can be disproven by a simple act of walking through a door" is clearly a reference to the earlier discussion of Zeno's Paradox. While it is true that the impact of his work is important and debated to this day, there is plenty more to Greek science than Zeno (who was more a philosopher than a scientist or mathematician anyway). The Greeks were the first to observe electricity and magnetism, for example. They made great strides in medicine for their time. Pythagorus, in addition to his famous work with triangles, developed early music theory with his study of vibration versus string length. For crying out loud, they nearly invented calculus!
3) You characterize the proofs offered for the commonly accepted value for Pi as "unverified ancient geometry/trigonometry", a phrase that elicited an actual LOL from me when I read it. Unverified? The Greeks built temples, some of which stand today, based on that geometry! So has every civilization since, including our own. As brainsandgravy above has posted, you can very easily verify it for yourself, any school worthy of the title will asign its geometry and trigonometry students homework or classroom activities doing exacly that, and I again urge any undecided readers to go out and perform those tests for their own edification.