I am assuming that if I use a shutter of a smaller diameter than the lens, the properties of the lens will change (for example, it will become less fast), because of the smaller aperture ring behind it?

That is not necessarily the case. The physical size of the aperture is (almost?) never the same size as the objective lens element; because its physical size is not what matters. What matters is whether the aperture obstructs the path that the light must follow while being focused at the image plane. That is the aperture's apparent/effective size as seen/magnified by the objective element (which is called the entrance pupil).

I.e. the aperture is (almost?) always a physical restriction; but what actually matters is if it is an optical restriction or not.

If the aperture diameter appears to be the same size as the objective element (due to the magnification of it), there will be no optical restriction to the light travel, and the lens's f-ratio/maximum aperture will be determined by the diameter of the objective element itself (it cannot be any larger). And if the aperture appears smaller than the objective element, then the lens's f-ratio will be determined by its apparent diameter because it is smaller and it is an optical restriction.