Yeah it makes no sense for him to want to rehab before signing. A team that wants to sign him for his upside would be more than happy to give him ample recovery time, and staff support for his recovery. His injury is not the type that any team would care about - it's not like he will have to demonstrate that he's healthy in order to get an MLB contract. His statement that he'll take time off to get healthy before signing doesn't seem to make a lot of sense, unless his injury is significantly worse than is being let on.
However, you're definitely wrong about the bolded statement. There are MLB teams with infield holes that Brett Lawrie could slide into and represent a projected improvement upon. Whether the teams are disinterested for reasons not related to talent is another matter.