You are correct that the type the compiler creates for a lambda is allocated in-place, usually on the stack, and perform no heap-allocs. However if you pass it to a std:: function it will be _boxed_ and std::function _will_ heap alloc the space for it. This alloc is what's not customizable.
That's fair, but std::function is not specifically about lambdas (As Boost.Function, it predates them, in fact) - it's about wrapping an arbitrary callable in a way that allows erasing its type. Idiomatic C++ rarely uses that class - I don't think it's used anywhere else in the standard library, even though it has plenty of higher-order functions etc. Turns out that closures that can only be passed in and not returned are still plenty useful.
In my dayjob as a code-reviewer, I see it in code-bases a lot. Between the generic name and elevated status in the std namespace, it's a natural tool for developers to reach for, across experience-levels. I speculate that the boxing side-effects are not well understood, given how many times I have to lift them out of hot-loops (despite the many unverified claims that "oh, LLVM will inline and optimize that away, no worries, teehee").
In general, I have not observed a consensus for 'idiomatic' C++, even within a single project. I say this as someone who wishes there was, because my job would be a lot easier if dependencies were less heterogeneous :)
