Their intention was to avoid the public eye and not be suspected. But Claudia's suggestion directly placed them in the public eye.
What made her suggestion very clever was that it omitted the suspicious part.
Leonard also understood that no matter how trained a person was, they were ultimately still human. And humans tend to have this curiosity aspect embedded in their program.
Hence, the spies sent by the wealthy and influential people were destined to forget about their presence even if spotted. After all, they were probably the slacker types as well.
Rather than sneaking around and even taking a large detour, which would involuntarily make them suspicious, they might as well walk through the gate openly and confidently.
At the very least, the spies wouldn't notice anything was amiss due to their confidence. Only the guilty would behave and show a look of anxiety.
