What the OP seems to want is a solution consistent with his own (broadly late P&PT) assumptions, not lessons on alternative theories of syntax in which his problem may not arise (if it doesn't; we would have to look at that in some depth).
Although I know nothing about Modern Greek and I am not even familiar with the way(s) generative syntacticians have analysed Greek clause structure either, herebelow is a proposal that, in my view, solves the OP's problem.
Assuming that things in Greek are as the OP says (i.e., that o Janis sits in Spec VP, whereas the verb has ascended into Infl/Tense), and assuming the - now obsolete - theoretical framework he refers to (basically that of Barriers plus the VP-Internal Subject Hypothesis of late P&P Theory), in which Nominative Case is still 'assigned' by T, etc., there are ways to assign Nominative Case to a VP-internal subject in situ. The obvious one is via association of the VP-Internal subject with an empty category (possibly a 'dummy' or 'little pro', see below) in Spec T, under the assumption that, for Case (agreement, categorial, and thematic) purposes, both are links of the same (subject) 'chain'.
Note that in English existential sentences like There has been a student waiting for you in your office the whole morning, a student, which remains in Spec VP (compare with A student has been waiting for you in your office the whole morning, where it has ascended to Spec T), is said to receive nominative Case via association, within the same 'chain', with the 'dummy' there that occupies Spec T. Recall that, since Spec T is not an a-(rgument) nor a th-(ematic) position, when the subject does not ascend from Spec VP into it, it must be occupied by a 'dummy' there, and only 'dummy' there can locally 'receive' 'structural Case' from +Tense. That - plus satisfying the Projection Principle, and doing so by providing phonetic content, since English is not a 'pro-drop' language and needs something pronounceable in Spec T - is the only function of 'dummy' there. Then, so the standard explanation goes, the subject in situ satisfies the Case Filter because the 'dummy' there associated with it does receive structural Nominative Case, cannot 'absorb' it (because it is not an argument), and must leave it 'available' in the subject chain for the NP a student to 'absorb' it. All that is consistent with late P&PT.
Back to the Greek case the OP asks about, to solve his 'problem' it would suffice to claim that o Jannis is also part of a subject 'chain' that contains a higher link in Spec T. The difference would be that, Greek being a 'pro-drop' language, that 'link' would not need to have phonetic content.
Depending on which version of P&PT one subscribed to, that 'silent' higher chain-link might correspond to the 'little' pro assumed to 'fill' apparently empty subjects in other 'pro-drop' languages (Spanish, Italian) or be filled by a phonetically empty Greek correlate of English 'dummy' there.
In either case, such a pro/dummy-DP chain would not cause any unification trouble at all, because a) there is no conflicting PF content (which is a unification problem in the case of English there-DP chains!), b) there is a single (Agent) theta role (assigned to o Janis in Spec VP), c) there is a single Case (Nominative, structurally assigned in Spec T, but 'absorbed' by the overt DP o Janis in Spec VP), and d) there is a unique set of Agreement ('phi') features (here: person, gender, number) inherently provided by the overt DP o Janis in the lower chain link.
Thus, as far as I can see, such sentences are not problematic for late P&P Theory.