You could, and then it would be limited by available number of police officers, and time, and so the risk of, and potential scale of, abuse would be far lower.
The opportunity cost is often the difference between something being reasonable and nearly the same act becoming grossly invasive.
A real world example: Norway has traditionally had public tax lists where anyone could see everyones taxable income and tax paid. Before the internet this involved going to an office, and so the opportunity cost was high enough that most people never would without good reason (e.g. suspecting tax fraud and wanting to substantiate it before going to the police; or investigative journalism). When the internet arrived, the lists were soon made available online. Suddenly all the newspapers offered searchable interfaces where you could look up all your neighbours and friends in seconds.
What had been seen as a reasonable tradeoff between transparency of the tax system and transparency of society (though some would argue it was encouraging snitching...) became seen as grossly invasive, and the question was raised whether to shut off access entirely.
Instead restrictions were added to try to redress the balance, and now while you can still look it up, there are limits (500 views per month, unless you're registered as part of the press, which has special access), and the person you look up can see who has viewed their information if they themselves log in[1], which raises the barrier to just randomly snooping.
The same information has been available to the same people the whole time. All that has changed is how convenient accessing it is, and how likely it is that snooping at someones data might affect you, and the result has been wildly different perceived levels of invasiveness.
