That is an odd version of politics.
Conservatives are, if we are going by definition and not party, those that support traditions and the status quo. Essentially, the job of conservatives in the political sphere is to protect those institutions that are working. Liberals are the ones that want to push those institutions further or, in the more extreme circumstances, remove them entirely.
I would say that means in any functional society, and our society is quite functional, that conservatives should absolutely be the 'big tenet party' aka has broad appeal across society in general.
The fact that the republicans are not that is not indicative of conservatism but rather indicative of the republicans failure to actually be conservative.