I actually don't think Eisenhower would've stopped being a Republican, however, he'd stop voting for Republicans at some point in the late 00s/early 10s.
I think it's a key distinction. I don't see Dwight Eisenhower becoming a Democrat, but I do think he'd stop actively supporting GOP candidates and just refuse to comment publicly on politics. (He actually gave quite a lot of cold shoulder in 1964 and did a lot of really embarrassing "damning with faint praise" to Goldwater that year)
I largely agree, but Trump might have pushed him over the edge. Ike disliked boorish people to boot.