equation solving – Calculating and displaying intersection of cylinder and line


I am trying to compute and display the intersection of a line defined by two points and a cylinder centered around the $z$-axis defined by length and radius. So far I have

cyl = Cylinder[{{0, 0, -1}, {0, 0, 1}}, 1]
line = InfiniteLine[{{0, 0, 0}, {1, 0, 0}}]
pts = Solve[{x, y, z} ∈ cyl && {x, y, z} ∈ line, {x, y, z}, Reals]

But this returns y -> ConditionalExpression[0, -1 <= x <= 1], z ->
ConditionalExpression[0, -1 <= x <= 1]
instead of a single solution. Any hint why this is the case and how to display the solution with the intersection in a nice way?