Older libraries that are long abandoned by their maintainers usually don't have anything. At work we pretty much ignore all libraries that don't have types from somewhere these days, but that doesn't eliminate many things.
Having to write the definitions myself usually isn't a huge pain, but it's a big red flag showing that it's probably abandoned, and there's no community support around it anymore (or possibly ever).
It's rare for there to be a breaking change in the JS emit.
The TypeScript team work hard to preserve compatibility. Breaking changes are explicitly managed and communicated ahead of time. There is a concept of a breakage-budget per-release. This means if you stay up to date, the cost of each upgrade should not be huge. Orta and Nathan on the TypeScript team talk about this in this podcast episode: https://dev.to/devteam/devnews-s1e4-typescript-4-0-gitee-chr...
So "keeping up" is not too hard.
A useful example is Strict Null Checks which started to treat unadorned types as non-nullable (string versus string? versus string | null). In most cases the runtime behavior of libraries were already throwing errors when nulls were passed in, and the extra checks helped immensely. In rarer cases "oh yeah, null is totally a valid thing to send to this API and it has some defined behavior that we document/should document" and the DTS files were encouraged to annotate their types to include that. In the worst where libraries had such APIs but failed to update their DTS files, there were explicit casting workarounds in consuming projects (the awkward `null as string` and the so called "damn it" operator `possiblyNullString!`).
Also, for the cost of dealing with a breaking linter once and a while it helps enormously with upgrading javascript libraries provided they have up to date type definitions. If an API changes you are immediately alerted to many of the places where your previous assumption are now wrong.
It also completely changes the way you write code. Often time you can just go to the leaves of your application, make the change you want, and then just follow the compiler errors all the way back up to the top. When it compiles there is decent chance that it will run successfully first time.