Only in a few cases you can find a meaningful answer to a question asking "why there is something lacking in a language". Normally, there is simply no answer. For example, why did not Ancient Greek have e Locative case in its nominal declension? Probably because it coalesced with the Dative. But why didn't it have a Comitative case? There is no answer to this question, because in Proto-Indo-European there has never been a Comitative case, without any reason.
Let's see whether type your question belongs to.
Usually, in historical linguistics to explain some feature of a language amounts to find its origin in some proto-language. So, let's start from clarifying what we mean by "stem" in the IE languages. The average IE noun can be either a radical noun (where case endings are attached directly to the root), or a thematic noun (where between the root and the case endings intervenes the so-called thematic vowel), or a noun that has a stem proper, i.e. a root plus a derivative suffix, with case endings attached directly to this stem. Now, let's exclude radical nouns and thematic declension and concentrate on nouns with a stem. Then your question must be reformulated: "why is there no nominal derivative suffix having the form of, or ending in, an -i diphthong?".
You are mentioning stems in -εύς as an example of diphthong ending stems. Now, let's exclude such words as βασιλεύς, Ἀχιλλεύς and other non-IE nouns whose origin is simply unknown and concentrate on words indicating professions: κεραμεύς 'potter' or ἁλιεύς 'sailor, fisher'. These stems are built with the suffix *ēw, which eventually evolved into something that was spelled, in Classical Greek, with the so-called diphthong ευ. But what is a diphthong? Phonologically a diphthong is hardly distinguishable from a mere sequence of a vowel plus a sonorant. In the teaching of Greek the notion of diphthong is widely used for the seek of simplicity, but it's more about spelling than about language. Suffice it to say, the "diphthongs" ου and ει are often just conventional spellings for long closed vowels /oː/ and /eː/ (outcomes of contraction).
All put together, you question can be reformulated in the following manner:
Why was there no PIE nominal suffix of the type -Vy- (where "V" indicates a vowel and "y" indicates a palatal sonorant), which then would evolve into something that is conventionally called "diphthong" in the Classical Greek spelling tradition?
Now, let's remember the PIE vocalic alternations: roughly speaking, every vowel could change into a zero. In the case of the suffix we are looking for this would result into a variant form -i-. Therefore, what we really need is a declension type with a stem ending in -Vy- in full (i.e. non zero) grade, preserved as such down to the Classical period. Unfortunately, there is no such thing.
- We have words with the zero grade of the suffix -Vy-, i.e. the -ι declension (πόλις). It does not show any "diphthong" (with the possible exception of some oblique case forms).
- We also have some words exhibiting the full grade of the suffix -Vy- (from IE *-oi-) but they have mostly lost the sonorant element, so no "diphthong" is there in the spelling. What I mean here is the declension in -ω, i.e. such words as ἠχώ 'echo'. The original Proto-Greek suffix is reconstructed as -οj- but the palatal element has been lost almost everywhere except for the dative form: ἡχοῖ.
References: Chantraine, Morphologie historique du grecque, §§82–88.
-es-ending stems" I actually mean words like Sōkratēs (Σωκράτης); the stem of this noun ends in -εσ, it's just the σ drops out when between vowels. So we have e.g. the genitive singular Sōkratous < Sōkrate-os < Sōkrates-os, where e-o regularly contract to ou in Attic Greek. Basileus, on the other hand, has the s just as a nominative singular suffix: g.sg. basileōs < basilē-os < basilēu-os; the -eu is nearly always -ēu, which is kind of confusing.
-e, if anyone reading is wondering. The vocative singular, supposedly bare, has -e, but everything else has -o. This is actually exactly like the situation of -eu nouns (-eu in voc. sg., -ēu everywhere else), except here most people and books think it is -o stem.