When are serverless functions overkill for a simple API?

I've been thinking a lot about serverless, especially Lambda, for backend APIs. For anything with complex workflows or event-driven stuff, it's totally a clear win. But what if you just need a simple CRUD API? Is there a point where all the hassle of dealing with Lambdas, even if you use a framework like Serverless, and those annoying cold starts, just makes it less attractive than a good old containerized service for a few basic endpoints? I sometimes wonder if we just jump to serverless because its new and exciting, not because it's always the best tool for the job. What have you seen?