There is no arms race. 99.9% of the time Google knows if you're a robot based on your browser state. They make you label the images because it's a free way to get training data.
If I'm trying to automate a system to fool their captcha, I'm probably getting a lot of bad results. Or I could just be intentionally feeding them bad data, the fact that not being allowed through captcha keeps letting me make more and more inputs would enable someone to do that as long as they would like to.
I don't know, maybe I'm missing something.
The thing you're missing is volume. Even if you assume the vast majority of people will attempt to mess up your data, when you have enough people doing it, you can look at them on aggregate and based on patterns disregard bad data. It might be "expensive", but still worth it.
Of course, if you run IRC bots that scrape Google with a headless browser to implement a .search functionality, and which offer link titling in IRC, and you use a separate bucket of cookies and IP for every IRC bot, your bots now also have a human search and browsing history, and also will pass all ReCaptchas...