Question

Why is using a compass and straightedge and compass better than technology

Answer 1

Technology comes from the Greek root $\tau \epsilon \chi$, meaning art or craft.

For the Greeks, a straightedge and compass was technology.

The nice thing about a straightedge and compass construction of any length is that there's always a corresponding algebraic form consisting of natural numbers combined via addition, subtraction, multiplication, division and square rooting (of positive numbers). So we get an "exact" answer, at least using radicals.

Compare that to the typical calculating technology we use today where the square roots turn into approximations. The calculator is worse, turning an exact answer into an approximation.

Straightedge and compass constructions play a large role in the development of mathematics but they're not really better, it's just how things went. By restricting ourselves to straightedges (linear equations) and compasses (circles, quadratic equations) we restricted the possible lengths we could construct. Understanding exactly how propelled mathematics forward for a couple of thousand years.