I don't think that's the case. Higher-order functions have been part of the OO toolbox since the batch of OO languages has existed, generally under the label "blocks" (indicating non-local returns separating them from the preexisting "lambdas").

And I'd argue that a language without anonymous first-class function can not be "truly" object-oriented, or at least not without significant syntactic overhead.

To me, a core idea of OO is that objects are opaque boxes to which you "give order" (that makes Erlang's processes excellent objects, by the way). But it is very common that you need to customize the exact behavior in some way.

Without first-class functions, the choices are to either extract all relevant data from the object, perform the operation yourself and (optionally) feed the data back, low overhead but a far cry from object orientation; or to create a new type holding all relevant customization information (that's the delegate pattern), very high overhead but remains object oriented.

First-class functions in OO language act as very-low-overhead delegates (you may still need full-blow delegates for complex customization, but single-method delegates aren't needed anymore), and let language users avoid imperative programming.