Let’s clear the air regarding the abilities of the Ancient Egyptians.
I only show this for the unit circle enclosed within G2's 3: 4: 5 triangle as the unit circle can be ascribed to represent G1.
The following is a closer examination of the unit circle with circumscribing square.
If the radius of a circle is one, the perimeter a circumscribing square has a perimeter value of 8 radii. Then 8 x r / (14/11) = 6 + 1/4 + 1/28 x r or 44/7 x r as the circumference of a circle with a radius of 1 or unit a circle. From my perspective it is evident the Ancient Egyptians were not cognizant of pi anymore than they were aware of phi, but instead found the 5 1/2 seked fulfilled both requirements in their world of unit fractions and rational numbers.
When I mentioned the stick system, it was to offer that the Sun can keep track of measure in distance and time, and by a measure which was mostly had by water dripping from a hole in a pot or even a heart beat at rest, time can be measured. If time can be measured, then where the sun is over a particular stick and the distance between sticks can be measured. When that distance is known and the length of a perfect day is known, the size of Earth can be known. If the size of Earth can be known, then the simple equation or rotation speed and distance and time offers up another way of estimating what pi is...but presented as something else instead of pi.
Not to rain on your parade, but the only catch to that, I can see, is what method of time keeping did they have that was accurate enough to make these calculations? According to all available records time keeping was not recorded Dynasty IV in units minute enough to make these calculations. And given the earliest surviving sundial dates back to 1500 BCE a thousand years after G1. So you have to ask and then demonstrate a sundial, or whatever method you can determine they used, is accurate enough to make these calculations?
Answering that question to a satisfactory conclusion turns a hypothesis into a theory worthy of further investigation.