The 'Elbow method' [1], yeah. Sweep a bunch of different k values and pick the one where your target metric starts having diminishing returns. There's still not a particularly great way to pick k for this type of clustering.

There kind of can't be as clustering is sort of inherently subjective. Other methods like HDBSCAN can find the number of clusters themselves. There's still a 'one key parameter' you have to pick yourself, but it can be more intuitive and less sensitive for some datasets, when you dont' know how many clusters there are / how many you want.

Sometimes, picking k exactly is useful for practical reasons too. Like bounding the size of your codebook. Or partitioning a dataset into k equal-ish pieces, say to spread across a pool of servers of a known/fixed size.

[1] https://en.wikipedia.org/wiki/Elbow_method_(clustering)