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I have used mixpanel and amplitude for product analytics, you could do in GA4 too, you just have to configure events on the activities

Looking for competitor traffic data? Similarweb, Semrush can show this. Available in paid plans.

Looking for own data, just more structured. I think events might be the way

Events are a good starting point, but for more specific analytics like what exactly was searched and all, you may need to go to backend as well. GA4 should be able to do this, but not exactly easy to set for a task like this

Something like this can help too maybe: https://www.algolia.com/products/search-and-discovery/analytics/