Planes can fly with the rudder inoperable, although with some restrictions -- you wouldn't want to do a serious crosswind landing, and you wouldn't want to stack it up with other failures, especially asymmetric engine failures.
However, that doesn't mean that planes can fly with the rudder /loose/. A significant risk in higher-speed airplane designs is that of aerodynamic flutter, where aerodynamic forces excite a vibration mode in the airframe, or a subset of it. You can find some impressive video of e.g. bending modes in sailplane wings being excited, with increasing magnitude bending until the wings are destroyed (or the excitation is reduced dramatically, or shifted to a different frequency). While aeroelastic modes get a lot of attention in flutter analysis, loose control surfaces can be much, much worse, because movement of the surface within the lash provided by the loose connections is effectively undamped.
However, that doesn't mean that planes can fly with the rudder /loose/. A significant risk in higher-speed airplane designs is that of aerodynamic flutter, where aerodynamic forces excite a vibration mode in the airframe, or a subset of it. You can find some impressive video of e.g. bending modes in sailplane wings being excited, with increasing magnitude bending until the wings are destroyed (or the excitation is reduced dramatically, or shifted to a different frequency). While aeroelastic modes get a lot of attention in flutter analysis, loose control surfaces can be much, much worse, because movement of the surface within the lash provided by the loose connections is effectively undamped.