> The warrant is the document the absence of which lets the public know

Er, the warrant is still there to be examined later, no? It's just not necessarily shown to the subject at the time of investigation.

Hence my second paragraph. “Don’t worry, we have a warrant” leaves the public vulnerable to misconduct, actions that potentially cannot be reversed or sufficiently compensated.

A warrant usually isn't a free pass to search everything. They are often narrow.

The warrant is the receipt. Even if you believe it's fine most of the time I'm pretty certain most people would feel uncomfortable if they went to the grocery store and weren't offered one. You throw it away most of the time, but have you never needed it? Mistakes happen.

The stakes are a lot higher here. The cost of mistakes are higher. The incentives for abuse are higher. The cost of abuse is lower.

And what's the downside of the person being searched having the warrant? Why does it need to be secret?

How can you be sure, when no one ever knows it is there to examine it?