This is a very good move. It would have been a very good move even if Lynn hadn't gotten hurt.
A lot of people seem to have missed the fact that the Sox aren't on the hook for the full $4.5M. His salary is prorated only for the time he spends in the majors. So basically $25/K per day he's on the active roster.
Nobody is expecting him to pitch a full season. If they get 10 5-inning starts out of him at a 4.5 ERA, that's a big help. That's fewer starts that someone like Lambert has to come up and make, or fewer starts that Lopez has to make which keeps him in the pen and makes it stronger, or, in a best case scenario, it's fewer innings that Keuchel has to pitch when nobody expects him to even be as good as a 4.5 ERA.
I still think they should have pulled the trigger on Manaea or one of the other starters who got traded, but this is probably the next best thing.