Hundreds of thousands (probably millions) of 4473s are sitting in storage containers outside of ATF. They don't have the manpower to search through them effectively or organize them to generate a paper database, let alone a digital one. There are apparently efforts to digitize them, but the digital files are (supposedly) not searchable or indexable (against federal law to do so). Regardless, digital files are easier to destroy in many respects. As it is, 4473s get sent to the ATF or destroyed regularly. So, I'm not worried about the 4473 functioning as de facto registration.
Besides, I have a concealed handgun license issued by a state. ANYONE can actually find that out with the proper request to the proper state government. So, it would be completely reasonable to assume because I have a concealed handgun license, I might actually own a handgun. That's not a big leap. So, I'm not really hiding.