There are numerous kinds of coordinate structures that challenge such an approach to coordination in terms of types. The main difficulty concerns the fact that the strings that can be coordinated need not be phrases/constituents, which means it is very difficult to assign any sort of type to them. An unstated premise in the question is, namely, that the conjuncts of coordinate structures are necessarily units of syntax, i.e. phrases/constituents, to which one can assign types. There are many cases where doing this is not possible. What follows is a brief, incomplete inventory of the kinds of coordinate structures that challenge theories of syntax based on semantic types, X-bar theory, or any other approach to syntax that deems the phrase/constituent to be the basic (and only) unit of syntactic analysis.
Syntactically distinct conjuncts:
When the coordinated strings occur as (part of) the predicate, they need not match at all in syntactic category. The next examples are taken from the Wikipedia article on coordination (https://en.wikipedia.org/wiki/Coordination_(linguistics)):
(1) Sarah is [a CEO] and [proud of her job]. - NP + AP
(2) Is Jim [conservative] and [a closet Republican]? - A + NP
(3) Bill is [in trouble] and [trying to come up with an excuse]. - PP + VP
(4) Sam works [evenings] and [on weekends]. - Adv + PP
(5) They are leaving [due to the weather] and [because they want to save money]. - PP + Clause
It should be apparent that some sort of super-type would be needed to accommodate these cases.
Right node raising:
The phenomenon known as right node raising (RNR) is such that the coordinated strings are clearly not constituents on the surface. The next examples are from the Wikipedia article on RNR (https://en.wikipedia.org/wiki/Right_node_raising):
(6) [Fred prepares] and [Susan eats] the food.
(7) [Larry has promised] but [Jim refuses] to support reform.
(8) [Jim can] but [Jerry cannot] make the meeting.
(9) [When did he] and [why did he] suffer a setback.
(10) [Sometimes she carefully reads] and [at other times she merely skims] the report they produce.
It should be apparent that it would be impossible to acknowledge the conjuncts indicated in these examples as constituents to which one could assign types. At the very least, the analysis needs to be augmented with some notion of ellipsis.
Non-constituent conjuncts (NCC):
The cases above could perhaps be rectified in terms of some sort of super-type and/or some approach that acknowledges ellipsis. There are other cases, however, where these additions are less plausible. The following coordinate structures are such that the intonation contour is normal (unlike in cases of RNR):
(11) Sam bought [wine yesterday in the local store] and [beer today in the big supermarket].
(12) Sam bought wine [yesterday in the local store] and [today in the big supermarket].
(13) Sam bought [wine yesterday] and [beer today] in the local store.
Most theories of syntax do not view the conjuncts marked in these examples as constituents. It would therefore be very difficult to assign any semantic type to them. The following further examples of NCC are taken from Wikipedia (https://en.wikipedia.org/wiki/Coordination_(linguistics)):
(14) [Did he] or [did he not] do that?
(15) She [has] or [has not] understood the task.
(16) Susan [asked you] but [forced me] to read the book on syntax.
(17) Jill [has been promising] and [is now actually trying] to solve the problem.
(18) [The old] and [the new] submarines submerged side-by-side.
(19) [Before the first] and [after the second] presentation, there will be coffee.
(20) Fred sent [Uncle Willy chocolates] and [Aunt Samantha earrings].
(21) We expect [Connor to laugh] and [Jill to cry].
Some of these examples might be construed as instances of RNR and could hence be grouped to the examples of RNR further above. In any case, they all help illustrate the challenge posed by NCC to any theory of coordination that assumes that the conjuncts of coordinate structures necessarily qualify as phrases/constituents.
In sum, the approaches to coordination suggested in the question (in terms of types) and in Lemontree’s answer (in terms of X-bar theory) are hopelessly outmatched by the data. Note that the challenging cases can be expanded if examples of gapping (https://en.wikipedia.org/wiki/Gapping) and stripping (https://en.wikipedia.org/wiki/Stripping_(linguistics)) are included. My personal stance is that to be plausible, the theory of coordination should eject the premise that coordination operates on constituents and allow it to operate on strings instead, whereby many of the relevant strings are non-constituents.
