Except that's not the calculus. By integrating chatgpt with their products, Microsoft pulls users away from: search, docs/workspace, and even gmail, and brings them over into: Bing, Office365, and outlook. In doing so, they threaten google's core income stream, and therefore google's ability to fight back.

Microsoft is not paying for an expensive search engine: Microsoft is paying (with chatgpt compute infrastructure) for a much larger piece of the productivity suite market, and hamstringing google in the process.