The OED describes had rather as the past subjunctive, meaning ‘would have’, and used idiomatically with adjectives (or adverbs) in the comparative, ‘to express preference or comparative desirability’.
Not all grammarians would now agree with the description ‘past subjunctive’ but the had rather construction is still found in British English. The British National Corpus yields 21 examples, but some are false positives. The Corpus of Contemporary American English has 23 records, but, as it is four and a half times bigger than the BNC, the incidence is relatively smaller.
It is no doubt true that the difference is often fudged by the contraction, as when Simon
and Garfunkel sang in ‘El Condor Paso’:
I'd rather be a sparrow than a snail