Harry Truman is a good example. The left-wing Dems hated him because he wasn't Henry (socialist) Wallace and because he wouldn't cave to the Soviets. The right-wing Republicans hated him because he wouldn't roll back the New Deal. The public liked him, and showed it at the voting booth.
What conservatives forget is that the president is president of all the people, and that includes the 50% who favor abortions; the 60% who want "better schools" (and likely think that the federal government has a role in them). I don't think it possibly to be ideologically pure and build any kind of support whatsover for more than one or two issues. Our system isn't made that way.
Jefferson caved on "free trade" and endorsed Gallatin's massive $10 million internal improvements plan (more than the entire fed budget!!!); Jackson trampled on the "states' rights" he endorsed in the campaign, and ignored the contracts negotiated by Indians; Teddy Roosevelt, in the name of "fair" competition, launched more anti-business suits than any other president. They were all enormously popular and (by all accounts) successful. Heck, even WASHINGTON enacted a massive tax increase, then sent the army out to collect it!
CFR should, and probably does alarm the informed electorate (those pesky extremists). Signing that bill was worse than clinton lying under oath.