Agreed – I’d say “ad business” instead of “search monopoly”: Chrome is easily justified simply by making sure that most users spend most of their time on web sites rather than native apps. Web users are heavily accustomed to everything being subsidized by ads, and Google is definitely winning there even if you rarely use search.
Android and Chrome make sense in terms of defending all of their web-based offerings from a hostile mobile OS or browser monopoly.
Their advertising dominance obviously is the only reason they had the money to throw at those problems, but a “search monopoly” isn't the only reason it made sense.