You still need to have one specific circle that you want to make the unit of measure because if you have 2 different circles, they will have different measurements.

To get a pi ratio, you need one measurement to be made with the diameter of the wheel, and another made with the circumference of the same wheel. You can certainly use ropes, but one of those ropes needs to be a multiple of the diameter, and the other a multiple of the circumference.

You might be measuring short lengths with a particular unit, say, a "stick". To get consistent longer measurements, you might measure 100 turns of rope around a spool one "stick" in diameter. The length of that rope might be called a "string".

The architects of a building might pound two rods into the ground, one "string" apart, and tell the masons to construct a wall between those rods. The architect wants a wall; the architect doesn't particularly care how long the bricks are. The masons don't particularly care how long the wall is going to be, just where they need to start and stop. The brickyard workers don't care how long a string is, they just need a consistent measurement for their molds.

Nobody involved particularly cares about pi, and yet the resulting building will have pi ratios all over the place.

Sure but wouldn't a wheel be easier?