That button gives a false sense of security, there are so many things wrong with it, that it's existence may actually harm users instead of protecting them.

• Browser history is persisted
• Screen does not immediately clear on the next rendered frame, DOM persists until google responds. If your wifi suddenly goes slow, it could be on your screen until the request timeout, ~10 seconds later!
• The hyperlink hitbox is smaller then the graphic. Might not exit, even if you clicked it!

Instead, educate users how to protect themselves:

• Setup an innocent looking "workspace" or "virtual desktop" bound to a hotkey. Most modern desktop environments come with this feature.
• Setup a disposable workspace with browser tab in incognito / privacy mode.

Advanced setup:

• use encrypted connection behind some kind of vpn, tor, proxy or gateway (https alone doesn't hide which website domains you have connected to)
• don't expose any personally identifiable information.
• persist nothing to disk
• kill all processes on leaving workspace
• sandbox processes

Tinfoil setup:

• Only use any software, firmware & hardware that you have compiled, installed, fabricated & verified yourself.
• Buy lots of aluminum foil and go full tinman.