There are many ways to construct an aperiodic set of points with algebraic coordinates, or even rational, or integer, coordinates.
Penrose tilings have vertices with algebraic coordinates. The new aperiodic "hat" tiling does too.
In fact, the hat tiling has an underlying periodic sub-tiling of kites, and you can use this fact to distort the hat tiling into an aperiodic tiling with integer coordinates.
The grain of truth in your argument is that algebraic numbers are more structured. Crucially, they can always be represented exactly with a finite amount of data. The Penrose vertex coordinates are particularly nice because they're rational linear combinations of 1 and sqrt(5), iirc, so you can represent them as a pair of pairs of integers.
(Edit: I'm also pretty certain the slopes of the cut-and-project plane are algebraic. You can obtain a nice one-dimensional aperiodic tiling by cutting a 2D grid at a slope of the golden ratio, which is obviously algebraic.)