The reality is simple, even if you do not want it, as a researcher you are something of a public figure. You are probably using public funds to do your research, you most probably train peoples (from undergrads to research assistants) and, sometimes, more than you think actually, you will be googled.
For all kinds of reasons you might not want to tell the world openly what kind of research you are doing (which is actually a shame) or even keep people for knowing your “at bat” scores (e.g. is your work actually being cited at all). Let me tell you a secret, unless you have never published anything, Google Scholar will find you… even if you do not want to.
So do not be shy and make your Google Scholar page public!