That's true, to a point. If Google's stock price fell off a cliff tomorrow it would be all over the news and their customers (advertisers) might have some questions. But there are two things to keep in mind:
1) If their stock price fell off a cliff, there's got to be a reason. It could be basically anything, but it has to exist and be well-known (maybe Larry Page had a breakdown, shaved his head and tried to run down some reporters, whatever). The event that caused stockholders to sell (thus crashing the stock) probably would have been enough to cause Google's customers to ask questions, the stock price is then just incidental. It's a "chicken and egg" problem in some sense.
2) Stock price really doesn't have any impact on the everyday functioning of an otherwise-healthy company. Stock price is a reflection of the buying and selling happening on the stock market, so fluctuations in price can actually have nothing to do with the current health of the company in question. Maybe tomorrow a report comes out that indicates search-based advertising will start to decline next year and will be half what is today in 2020. This would likely lead to a decrease in Google's stock price. But it probably wouldn't lead to many companies dumping Adsense, at least not yet. A report about the future affects the stock price, but it doesn't necessarily affect the customers today.
Stock prices are abstract and largely disconnected from the actual company. So while a good CEO will pay attention to the stock price, it just isn't all that important for most (especially mature) companies. Building a healthy business is the important part.