"We pick two locations on the same latitude".
Erathostenes used different latitude but same longitude to know Earth's circumference.
If you do the experiment with the same latitude you'll get as Oscar Lanzi said :
"the circumference ... of your latitude circle"
And thus to know Earth true circumference you'll need to do the maths trick ...
Short version; the beach moves because the waves change.
Long version; Beaches, and in fact entire bays, conform in shape and alignment to the prevailing wave patterns in the area. The exact magnitude and alignment of wave trains varies due to chaotically complex interactions between surface winds, subsurface currents, local seabed features, tide, larger ...