When does the Tumbler have a higher catch rate?

The condition to remember is narrow: a gather, during a nurture event. That is the only situation where the multiplier reaches 6x. The Tumbler still raises your success chances in other capture situations, but the size of the boost outside a nurture-event gather is not fixed at that number.

Practically, that makes the Tumbler an item you hold rather than spend. Using one on a routine open-world catch works, but you are spending the boost at its weakest. Save them for when a nurture event is running and a gather is on, because the same item is worth six times as much there.

For everything else, capture comes down to your Pods. Aniimo Pods come in different tiers and types, and each gives a different success rate, so the normal way to raise your odds on an ordinary catch is to bring a better Pod, not to burn a Tumbler.

Not settled: Which capture situations other than a nurture-event gather trigger the Tumbler's bonus, and how large that bonus is in each, is not pinned down.