Google's primary objective for Chrome and Android was simply to get people to use the internet more, on the (entirely reasonable) assumption that they'd probably use Google and they would see a lot of AdSense units. They invested heavily in getting better software into the hands of as many users as possible. Of course they harvest a whole bunch of user data, but dragging up the quality of the browsing experience was a far bigger factor IMO.
Would Edge exist if Chrome had never happened? Would Firefox have a fast JS engine and proper sandboxing? What would the non-Apple smartphone market look like without Android? Whether you like Google's business practices or not, they've massively increased our expectations of browser software, in the same way that Starbucks created a world where you can buy a half-decent cappuccino in McDonalds.
Google paid Apple 12 billion dollars last year to be the default search engine in Safari and another half billion to Firefox for the same. Google's primary objectives with Chrome was to lower that amount and to guard their monopoly on search.