I post this in response to Wojowu's request in a comment; I don't really consider this usefully addressing the question raised.
I thought of a simpler construction. Take a square with mirrored sides, and poke a point-hole in the interior of the top edge. When a ray from outside with an irrational slope (w.r.t. the box sides) enters through the hole, it is trapped. For if it escaped through the hole after $k$ reflections, then it would contradict the irrationality of the slope. It is known that the path of this "billiard" is aperiodic (a result due to G.A.Galperin, I believe).
[![RaysInSquare][1]][1]
And of course almost all rays have irrational slope.
But as soon as the point-hole is enlarged to a positive-width hole, the ray will eventually escape (Poincaré recurrence).