It seems like he be need to stay in the birdcage as center to controll it, would make sense overall.
Meaning he probably can´t use it outside of the birdcage, can´t controll it.
Also the thing is, Godthread is probably the strongest attack of Doffy, while awakening could be based on birdcage.
Meaning G4 attack should be able to break it, the sad part is that actually no one really try to cut it(plot, ask Oda, don´t know).
Alone the fact that a navy Admiral didn´t do something against the birdcage(he gamble and put anything to Luffy with meaning he probably not try it at much) showing how much plotarmor the birdcage had, to make a little tension.