Given that we say things such as "I'd rather (that) you do it.", I'd expect "I had better (that) you do it." to be possible as well to mean "I would consider/find/have it better that you do it.", analogous to "I would consider/find/have it rather that you do it." (rather being the comparative of the archaic rathe).
However, a quick search would show that this isn't used and is likely ungrammatical. Is there any reason behind this?
S
, but not for several hundred years. Anyway, would rather is an idiomatic paraphrase for prefer, and had better is an idiomatic paraphrase for should. As idioms, they are not compositional, and don't follow whatever rules you expect. Idioms make their own rules; they're frozen, like individual words.S
used to be possible. What may be idiom now might once have been governed by rules before. This question was a probe in that direction, however impractical it may be to analyze idioms.