Has there ever been a weaker Church-like thesis? Background. The Church-Turing thesis, in one of its many equivalent formulations, states that the intuitively computable arithmetical functions are exactly those computed by Turing machines.
According to Alan Turing’s classic paper On computable numbers, with an application to the Entscheidungsproblem, “intuitively computable” refers to a human computer having access to enough scratch paper to hold the intermediate results.
This thesis has been extremely successful among logicians first (including Kurt Gödel), and computer scientists later; some authors even extended it to include all functions that can be computed by any effectively realizable physical system.
Nonetheless, the Church-Turing thesis is, at least in principle, falsifiable: it is enough to describe a non Turing-computable function admitting another kind of computation procedure, executable by the above-mentioned human computer. Of course, no such function is known to exist; however, consider the following “weaker computability thesis” for the sake of argument:

Every intuitively computable arithmetical function is primitive recursive.

This is falsified by Ackermann's function, which is clearly computable (both intuitively and by a Turing machine) although not primitive recursive.
Question. Has a similar, provably weaker “computability thesis” ever been proposed before Church’s and Turing’s? As an alternative, can we reasonably argue that no such statement was ever made?
 A: Although your question is a historical one that really should be investigated by historical methods, there is an abstract argument that no such thesis was previously made.  Namely, at first glance it seems impossible that one could characterize the intuitively computable functions, because given any such precise definition, couldn't you just diagonalize out of it to get an intuitively computable function that is not in your original class?  For example, you can think of the Ackermann function as diagonalizing out of the primitive recursive functions.  Surely a similar trick would apply to any other proposal?  I seem to recall reading somewhere that even Goedel had this intuition at first.  Thus until the recursive functions emerged as a specific candidate, it seems unlikely that anybody would have been tempted to formulate a CT-like thesis for any other class of computable functions.
A: I think it unlikely that anyone ever proposed a weaker Church's thesis,
because, as Tim Chow points out, diagonalization was known (and known to be 
constructive) before anyone ever contemplated a definition of computability. 
As early as 1907, Brouwer observed mathematics seems to be incompletable 
because of diagonalization, and Goedel thought that there could be no formal
concept of computation until Turing's definition persuaded him otherwise
in 1936. He later said that it is a "kind of miracle" that computability
can be formalized while provability cannot. 
Also, Post arrived at a formal definition of computability, via his
concept of normal systems, in the early 1920s, though it was not published. 
So the full concept of computability actually arrived before weaker concepts
such as primitive recursive functions.
A: I think you can go further and say: "effectively computable" means computable in polynomial time.  These two articles might be of interest for that sort of viewpoint:
Scott Aaronson, NP-complete Problems and Physical Reality, ACM SIGACT News, Vol. 36, No. 1. (March 2005), pp. 30–52.  http://arxiv.org/abs/quant-ph/0502072 
Wigderson, Avi (2010), "The Gödel Phenomena in Mathematics: A Modern View", Kurt Gödel and the Foundations of Mathematics: Horizons of Truth, Cambridge University Press  http://www.math.ias.edu/~avi/BOOKS/Godel_Widgerson_Text.pdf
