The fact that the Washington Post chooses to treat Kristol as a "Conservative" leader, confirms his status with the Leftist media. It should be a red flag to any duped Conservative.
I have no idea who David French is; but if he is working with Kristol, he is not likely an American Conservative.
See Neocon Phenomenon, to understand how truly far from Conservative--or any shared values--are the Kristols.
This is not too hard to believe because Karl Marx praised the 19th century USA as the place he hoped the socialist revolution would take hold and flourish in a best case scenario.
The neocons did abandon planned economics but held onto the idea of the USA should be a big govt in terms of military planning and as a security state. The Military Industrial Complex form of socialism. This Military Industrial Complex is to be used to bring about a some sort of Pax Americana. The neocons don't like any notion of independent regional powers, even of they are not hostile to the USA and are democracies. All nations need to be brought to heel as client states of the neocon US empire in order to fulfill the globalist revolution.
There are other globalists out there like the UN globalists - the neocons are just one subgroup of this thinking.
The first battle to be fought is not left versus right but inside versus outside. The Kristols and the Medveds of the world are no less elitist than the Obamas and the Clintons.