Actually, 翅膀硬 is a set phrase literally meaning 'sturdy wings', so OP's initial guess was (nearly) correct. It is a complaint used by parents or guardians to describe their children or subordinates who start to have their own opinions and seek for independence, just as little birds who have just got full-fledged wings and try to fly away from the nest.
Note that when parents say this, it most probably means that they do not like or agree with or support their children's idea. If they do, they would usually say 你长大了, rather than 你翅膀硬了.
So the whole sentence means:
So you think you have finally grown up, do you? If you want to fly away, so be it then.
翅膀硬, which means literally "wings grown strong" or figuratively "ready to try/spread one's wings and leave the nest", is in some way similar to English full-fledged "having reached full development; ready to fly". As Stellari correctly pointed out, 翅膀硬 is an idiomatic expression typically said by a sponsor/fosterer/superior in reference to a dependent/underling that has become so bold as to have a mind of his/her own.