I have a friend who took his private-pilot exam in a rudderless Ercoupe and was restricted. (Un-be-knownst to him, Which caught up with him when he sold the 'coupe and bought a Tri-Pacer. But that's another story.)
That was a concern for me to the extent that I put off finishing up my training, for a couple of years, looking for the right plane that I could fly with hand-controls. (I'm Post-Polio) Unfortunately I was having no luck finding one in my area. Last fall a couple of aviation friends talked me into finishing up in the Ercoupe. Explaining that if I did get restricted, I could take care of it later. At least I'd have my ticket.
However, having passed the sport-pilot check ride a couple weeks back, I can tell you that I have not been restricted. And, according to my CFI, becasue of the "class" brackets in the LSA regs, they can't restrict you. Don;t know for sure if that's true or not. All I know is that I'm not restricted. (Which means I wasted 2 years, worryin' over nothin'. Typical me.)