Why does producer theory make the assumption it does, and what types of firms in which industries is producer theory most likely to be an accurate approximation to?
The standard model of competitive producer behavior "the producer theory" assumes that the firms are 'price takers', technology is 'exogenous' and that firms maximize the profits.
To rephrase the assumption a little, firms are assumed to have technologies that are fixed (not newly innovated) and exogenously given (not under their exclusive control) and that convert the inputs or labor, land, raw materials and capital into outputs of salable products. It is also assumed that production plans are chosen to maximize profit and that producers accept prices--both input prices and product output prices--as they exist in the marketplace, or as given: It is assumed that producers cannot individually affect the rise or fall or any change at all in the input or output prices (Levin and Milgrom, "ProducerTheory").
As to why these assumptions are made, that is a controversial question among economists who hold many differing opinions including the extremes of rejecting all assumptions or accepting them all unquestioningly. Levin and Milgrom say this about why assumptions are made:
All economic modeling abstracts from reality by making simplifying but untrue assumptions. Experience in economics and other fields shows that such assumptions models can serve useful purposes.
It is still possible to answer "why" by saying assumptions are used for three main reasons: support economic tractable models highlight certain economic effects; provide the basis of numerical calculations for formulating economic policy or estimating economic affects; to provide reasonably accurate predictions of affects in the economic environment.
While these assumptions have often been criticized for being too narrow or somewhat unrealistic, they do serve important purpose. Any model is a simplification of reality and so is this theory and it achieves this purpose by making simplistic assumption (may be untrue or non-inclusive of some factors) of the reality. One achievement in such a scenario is the isolation of important effects for analysis by suppressing other factors. Secondly, it enables quantification of various factors by numerical calculations, which can be used for policy formulation.
I would assume that firms working on a franchise model are best fit to producer theory. Think about the local Burger King or McDonald store. Technology is exogenous (given by the headquarters), prices are fixed and the franchise has to maximize the profits to sustain.
Similarly think of licensed drug manufacturing, where a company obtains the license from a pharmaceutical for production and sale. The firm owns the 'permit' to manufacture, but prices are market driven and technology is exogenous, yet the firm has to make profit.