Remember 99.9% of cyberware was developed for the medical commercial market: To be used on individuals that had suffered an accident or disease.
For the market they are designed for, wireless makes sense. They can have their 'ware monitored by DocWagon for signs of problems. They can get software
updates from the manufacters easily, and effortlessly.
Now, for the 1% of the market that goes "wireless is bad, mm'kay" <Shadowrunners, off grid workers, luddites> Well, they can turn it off and "lose out" on those nifty features... like auto updated software, cloud storage, and what not.
Now does that mean you couldn't store that video in the cybereye? No, it just means the video stays in the eye.
Now, as to how much can be stored in the eye... well, that's going to be up to the GM, because like I said before, 4e got rid of storage tracking altogether with their "you can store it anywhere" motto, and 5e expanded with "cloud" storage.
A few minutes to hours storage in the eye? Sure, why not? You're entire waking life since the eye install? Probably not.
So this leaves runners in a catch 22 situation. If they wonder about "wireless off" in the wrong areas, well that attracts attention. However, wondering about wireless on, does leave them open to attacks and tracking.
There is NO perfect solution, you don't get to have your cake, and eat it...
You have to manage the risks based on the situation you find yourself in. If you are in an AAA Zone, going wireless off is not just a giant neon sign that says "HEY! ABOUT TO DO SHADDY SHIT HERE!!" its against the law! (And thus, they are already after your ass). Going wireless off in the RedLight District? Well no one gives a shit. Hell its expected! So a Runner has to plan, and really think about what they are doing, and WHERE. In that AAA Zone, where MAD sensors, Chemsniffers, RFID readers are the norm, along with random security foot, drone and astral patrols: Do they really want to wander around not broadcasting? Do they really need 6 guns on their person? Are their licenses going to stand up to checks?
Now, some of this you can get around. Optical chip burners and chips to keep private data private. Good Fake SINs for cover, and licenses for most ware. And yes, even the ability to turn wireless on and off is an advantage. Wireless on, you are just a Stream of Data in an Ocean, and when the shooting starts, you go offline for the advantages wireless off gives you,cand deal with the clean up later...
But you're never going to have all the advantages of being online, while being wireless off... Everything has a price.