It's relatively easy to construct a scenario where more search is in fact indicative of better search. To stick with Spotify: let's imagine they have an amazing search tool that consistently finds new, interesting music that the user genuinely likes. I can imagine that in that situation, users are going to search more, because doing so consistently gets them new, enjoyable music.
But the opposite is equally possible: a terrible search tool could regularly fail to find what the user is looking for or produce music that they enjoy. In this situation, I can also imagine users searching more, because it takes more search effort to find something they like.
They key is why are users searching. In Spotify's case I imagine that you could try and connect number of searches per listen, or how often a search results in a listen and how often those listens result in a positive rating. There are probably more options, but there needs to be some way of connecting the amount of search with how the user feels about those search results.
And yeah, using nothing other than search volume is probably a bad way to go about it
Or more saves and thumbs up on signs resulting from a search is because users are desperate to save a song they like because they have no faith that they'll be able to find it again with search.
The only way is to use the product yourself and honestly engage with it. Stats can't answer this question.
I feel like understanding this difference is what a good product manager should be responsible for. Not just optimizing any metric that is available but understanding the meaning behind them and choosing the push them the right direction.
But the opposite is equally possible: a terrible search tool could regularly fail to find what the user is looking for or produce music that they enjoy. In this situation, I can also imagine users searching more, because it takes more search effort to find something they like.
They key is why are users searching. In Spotify's case I imagine that you could try and connect number of searches per listen, or how often a search results in a listen and how often those listens result in a positive rating. There are probably more options, but there needs to be some way of connecting the amount of search with how the user feels about those search results.
And yeah, using nothing other than search volume is probably a bad way to go about it