"Inability" is a bit of an overstatement, but you're right that it's distinctly nontrivial - both branches of the family underwent some large-scale changes in phonology, and Korean's first unambiguous attestation is comparatively late (Middle Korean written in hangul dates to the 15th century CE, with older attestations written in Chinese characters which are very much not straightforward to read).
The most recent work on reconstructing the common ancestor of Japonic and Korean that I know about is Alexander Francis-Ratte's 2016 dissertation. Here's an example of the difficulty in reconstructing a common ancestor for these languages, which Dr Francis-Ratte explained to me in more detail when I contacted them about it. The dissertation lists two possible etymologies for Old Japanese koto "different", but they're more confident in this one.
Old Japanese koto "different" : Middle Korean thul- "twist/turn something"; thulGí- "be wrong"; thúm "difference, gap"
thúm and thul- indicate an underlying *thu- via the addition of nominalizer -m and continuative -(o/u)l- morphemes respectively. Aspirated consonants in Middle Korean descend from crasis of *k or *h with a plain consonant, hence *kutu / *hutu, either of which could correspond to Old Japanese koto /kətə/ - the latter would be proto-Korean-Japanese *kɨtɨ, the former *xɨtɨ.