What exact role do higher-order combinators (or function producers) hold in concatenative and tacit programming?
Is there another way to implement a concatenative programming language rather than directly manipulating the stack?
How tight is the relation between the combinators mentioned and Schonfinkel's combinatory logic?
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Is there another way to implement a concatenative programming language rather than directly manipulating the stack?
Yes. The Enchilada language is based on term rewriting.
In my answer to "Explain Concatenative languages to me like I’m an 8 year old" I argued that it is not useful to use the word "concatenative" to describe programming languages. This area appears to be a private playground for Manfred von Thun. There is no real definition of what constitutes a concatenative language, and there is no mature theory underlying the idea of a concatenative language. Because the concept is not really defined, it is impossible to provide a definitive explanation of the role of combinators.
You might prefer to spend your time studying more well-established topics such as combinatory logic, SK combinators, lambda calculus, and graph-reduction machines. If you are interested in point-free programming from a more pragmatic perspective, APL, Forth, and PostScript are all well worth studying.