“Loath” is an adjective meaning “reluctant.”
“Loathe” is a verb meaning “abhor.”
Therefore a phrase such as “They would be loathe to admit it” is gibberish. Any given instance could be a mere typo, such as I constantly make in this space. And it’s a typo the spell-checker won’t catch, since “loathe” is a perfectly good word. But I detect a trend of substituting “loathe” for “loath,” which would be a shame.