There was virtually no time Taft & Ike had to work together. Taft transferred duties as Majority Leader when he prepared to go to the hospital, this less than 5 months after Ike was sworn in (June 1953) and died the following month. Bill Knowland stepped into the job of Taft, but LBJ was running the show before long, so there was no “Conservative” agenda to speak of.
And, again, whatever Ike revealed in private demonstrates nothing if it doesn’t translate to a similar agenda, the abortion example I gave you. Ike had a premier opportunity to move the nation back to the right, to drastically cut tax rates for job creators (AKA the rich), to weed out the Communist and Communist sympathizers within the government and our public institutions (for which McCarthy was demonstrably proven to be right about, as the Soviet Venona Documents confirmed). Instead, the “pragmatism” kept in place everything a quarter-century of Socialist expansionism had wrought since Hoover’s tinkerings (himself a left-winger Progressive Wilsonian Democrat who switched to the GOP before the 1920 elections solely out of opportunism as he knew the Dems would be obliterated that year and he didn’t want to be on the losing side).
As with Ike, Nixon/Ford, Bush Sr. & Jr., they pissed away Conservative opportunities willfully so, clearing the way for far worse Socialist (or worse) policies from their successors.
With Ike’s failure to address these Socialist (so-called) “safety net” policies and their viability for the long run, he did the nation a grave disservice and paved the way for expansionism upon such policies that have driven us to such obscene levels of spending and debt. Real leadership at the time would’ve seen the enactment of a balanced budget amendment, for starters, to assure future spending sprees would be made a prohibitive venture.
All in all, the Democrats and the left ultimately lost little and gained everything under Ike’s “leadership” (ditto Nixon and the execrable Bush Dynasty).
The conservative opportunities that you see for Eisenhower and Nixon were largely illusory. The New Deal and FDR were still popular, and the public had not yet had the experiences of the 1960s and 70s to make them ready to question liberalism. Only then did modern conservatism accrue enough strength enough to win, and even then, the extraordinary political talent of Ronald Reagan was required.
Moreover, in the 1950s, the intellectual case for tax cuts had not yet been developed, and, without the far greater burden of Medicare and other benefit expansions, Social Security was affordable and sustainable. And not until 1974 did Jude Wanninski arrange for economist Art Laffer to meet with Don Rumsfeld and Dick Cheney to demonstrate new economic findings by drawing the Laffer Curve to show how lower income tax rates could bring in more revenue than higher rates.
Insisting that Eisenhower as President should have arrived at such an insight in the 1950s unaided is like insisting that he could have defeated Hitler more quickly if only he had used cruise missiles and JDAMS delivered by B-2s. Sure, he could have, except that he didn't have any of them available because they had not been invented yet.