I think this is similar to the question "does using turn-by-turn GPS make me a better navigator?" The answer could go either way. If you depend on the GPS to the exclusion of paying any attention to your surroundings, you'll probably be a worse navigator, because it will take you much longer to learn even the common routes in your day-to-day life. If, on the other hand, you use GPS-aided navigation as an opportunity to learn the back roads that take you between two places more efficiently, but that you never would have discovered on your own, and internalize that knowledge, GPS can help you become much more intimately familiar with an area than you otherwise might be.

I see Google/Stack Overflow through the same lens. Lean too heavily on snippets of other peoples' code, and you'll never learn to solve even simple problems on your own. But if you take the time to actually learn why a particularly elegant answer on Stack Overflow works, and generalize that technique, you can grow your skillset much more quickly than by trying to figure out everything yourself from first principles.