Can we derive Gk ἐυπάτωρ 'having a good father' straight from an IE root? The root would be a so-called nominal composite, that is, a noun formed from two elements, in this case an adverb and a noun. The first element of the compound is unproblematic *h1esu; it is the second part that may cause some troubles. The second nominal element is obviously derived from *ph2-tér-s (nom. sing), which is a standard hysterkinetic noun. The question becomes, then, can we derive an amphikinetic noun from a hysterokinetic one? The second element of the nominal composite would have to be *pá-tōr-s. The accent shift is interesting. From the root *ph2-tér-s, two methods of derivation are possible. First, the root with a syllabic laryngeal would have to hold the accent, so we would have *ph2-tōr-s. There are apparently roots which have accented syllabic resonants, but the roots (the ones for 'wolf' and 'bear') are a bit sketchy to being with, and to my knowledge there are no roots which have an accented syllabic laryngeal. So this, to me, seems unlikely. The second possibility is then the best one: namely that the accent shift occured after the 'loss' of the laryngeal and the creation of the 'a' vowel. Thus, in my view, we would have
*ph2-tér-s > *pa-tér-s > *h1esu-pá-tōr-s >ἐυ-πάτωρ.
*ph2-tér-s > *ph2-tōr-s [with the accent on the root] > *h1esu-pá-tōr-s >ἐυ-πάτωρ.
Given all of this, it seems that whether ἐυπάτωρ comes from a 'true' IE root hinges upon when the shift from *h2 > α happened, which presumably was a post-IE-pre-Greek development. Thus, I would argue that the Greek word cannot be derived straight from an IE root, given that the shift from *h2 > a is a prerequisite of the derivation of the nominal element in the compound. To be fair, though, nominal composites such as this need much more work. Perhaps it's a future project for me.