Actually there are a lot of cases where side-channel are important threats.
To name a few:
- IOT devices that are storing secret keys to communicate to each other or do whatever they need to do with it, and that are not supposed to be impersonated by an attacked from his computer.
- Trusted Execution Environment such as TrustZone, SGX or TPMs in general. For a very recent example, see tpm.fail.
- cloud computing: being able to read somebody's else secrets in memory because you are both sharing the same server in the cloud's system
- Pay-TV systems or any other system where the providers is broadcasting encrypted content, that is meant to only be decrypted by authorized key holders. (Typically done using smart-cards that are supposed to be impossible to clone, and whose key should be impossible to extract.)