The first two pages make the most of the built-in index. Mixed result sets gives pages, posts and products one shape and a sensible order in a single list. Tuning relevance lists the levers a view offers, the ones it does not, and a method for tuning against the queries people really type; read it before promising a client "better results".
When to use an external search engine sets out the limits of the built-in index and the factors that justify a second engine, without recommending either. Keeping an external index in sync is for after that decision: a job that pushes each change from record events, a scheduled job that corrects what the push missed, and the monitoring both need.