Marines would be obvious choice, and I think the significantly expanded harbor facilities would also indicate the possibility of a significant naval presence. They may even base a few of the new 056s there in a few years after they start coming online in numbers.
The airport facilities have been expanded, but not by much. Woody island lies in typhoon territory, meaning if you want to permanently base fighters there, you need to construct hardened shelters to protect the fighters from periodic storms and also from te elements. That would be particular important here because of Typhoons and also the close proximity of the run way to the waters edge. Leave fighters out in the open there and corrosion will get them even if a storm doesn't.
The island can and will be used as a refuelling and rearming post for fighters, and would greatly expand their range and endurance and fighters could easily be forward based there for short periods if tensions spike. But right now, there is simply no need to permanently deploy fighters there, nor have the necessary preparations been done to allow that to happen. Deploying fighters there would only serve to needlessly heighten tensions, and would actually place those fighters at greater risk as they would be far closer to potential enemies; the airfield would not be protect anywhere like as one on the mainland; and unless they build a big new radar station on the island, they would not have anywhere close to the warning time they would have on the mainland if someone decided to launch a surprise attack.
Fighters are just not needed there at present and would be a waste and placed unnecessarily in harms way if you do base them there. If the island is ever to get a permanent aviation detachment, helicopters would be far more useful and cause far less tensions.
