In Indo-European studies, there's always talk of active-stative languages, as an attempt to unite the Anatolian mi-class/hi-class dichotomy with the trifurcated present-aorist-perfective system preserved in non-Anatolian dialects. Somehow this hasn't completely sat well with me and I'm starting to understand why that is. For instance, concerning the absence of a verb 'to have' in Indo-European, Brigitte Bauer in Archaic Syntax in Indo-European first goes on to explain: "Instead, [Proto-IE] had constructions of the type mihi est, which includes a stative verb 'be' [...]" Notice that she rightly identifies 'to be' as a stative verb here. Yet before listing off some examples of active-stative verb pairs, she shares in the same paragraph: "In addition, Indo-European languages display residues of a lexical distinction between active and stative verbs. There is a group of verbs, moreover, that have two lexical roots, both referring to the same verbal concept, yet one representing its active aspect, and one representing its stative aspect."
So what's the problem? The problem is that the first pair she lists is *h₁es- 'to be' & *bʰeuh₂- 'to become' and if it's true that one is "active" and one is "stative" in a system where the active verbs are supposed to be marked by the *mi-set of pronominal endings and the stative verbs are marked by the *h₂e-set, then she and other Indoeuropeanists appear to have contradicted themselves. Further, there's the whole matter of *-i supposedly marking the "present tense" that I think I'm going to take up issue with right now.
According to the standard view, heavily biased in favour of non-Anatolian PIE languages and an effectively post-PIE state of affairs, we have the following forms and their functions:
bʰér-e-ti '(S)he is carrying' (present)
bʰér-e-t '(S)he was carrying' (preterite)
bʰēr-s-t '(S)he carried' (aorist)
bʰé-bʰor-e '(S)he has carried' (perfect)
This is not acceptable as a genuine model for the Proto-Indo-European stage because it doesn't explain a lot of things about the oldest branch of them all, Anatolian, and its unique features like 1) its scarcity of thematic verbs in *-e-, and 2) its system of mi-class and hi-class conjugations that are kept separate. Also, if the above is correct, we have an ugly situation where the present appears to be marked with an extra ending *-i despite a strong tendency in languages for presentives to be less marked than their preterite counterparts. So let's remodel the above into a new, purely aspectual system which might better account for everything, including Anatolian this time:
bʰēr-t '(S)he carries/(S)he carried (habitually)' (non-continuous)
bʰḗr-ti '(S)he is carrying/(S)he was carrying' (continuous)
bʰēr-s-t '(S)he carried (once)' (experiential)
wóid-e '(S)he knows/(S)he knew' (punctive)
I've tossed out thematic verbs which have already been nicely explained away as derivatives of subjunctive forms. So *bʰer- 'to carry' is now recognized as a former Narten present. I've also made the so-called "preterite" the new default, specifying non-continuous (ie. completed) actions or states regardless of whether they were contextually meant to be past, present or future tense whereby the form may just as well have meant '(S)he carried' (past) as '(S)he will carry' (future). The so-called presentive is reinterpreted as a continuous action or state, again regardless of time reference. Finally the sigmatic aorist is no longer treated as a formalized conjugation distinct from the non-continuous. Rather, the marker *-s- specifies a specific event of an inherently dynamic verb (a perfective nuance) and this is why only some verbs were given a sigmatic aorist later on in non-Anatolian dialects.
My understanding of PIE grammar is being inspired by Mandarin Chinese where here too, tense is irrelevant. Both the non-continuous and punctive would be expressed in Mandarin with the perfective particle le (了) placed after the verb. Continuous is very similar to the preverbal progressive marker zài (在), except that the PIE marker also marked stative *h₁es- 'to be' while Mandarin shì (是) isn't marked this way. The experiential is reflected as the particle guo (过) and already, by speaking of an action experienced, is inherently preterite to begin with. More on the fascinating subject of Chinese aspects can be read here if one wants to explore the implications of a tenseless language further.
I notice that le is often mistaken by foreigners (like me) as a past tense marker, yet it's more accurately described as a marker of completion for both action and state, refered to as a perfective or completive. The fact that this marker is mistaken so often as a past tense marker by those more familiar with tensual languages however suggests a strong semantic connection between the two. The common "completive" nuance between the two concepts provides the conduit in Indo-European for the eventual restriction of non-continuous aspect to past tense (even in Anatolian) and the reinterpretation of the continuous in *-i as a present tense which spreads to the Anatolian hi-conjugation as such. All this being said, we then understand why the perfective action could not possibly have been originally marked by *-i (as proven by non-Anatolian dialects) if its function were originally to express this aspect, due to the obvious semantic contradictions that would ensue, and we also see why the sigmatic aorist couldn't have ever applied to all verbs, such as punctives, likewise to avoid simple contradiction.
 Bauer, Archaic syntax in Indo-European: The spread of transitivity in Latin and French (2000), in Trends in Linguistics 125, p.21 (see link).
 Lehmann, Theoretical bases of Indo-European linguistics (1996), p.219 (see link).
 Bybee/Pagliuca/Perkins, The evolution of grammar (1994), p.140 (see link): "For instance, progressives that definitely are not restricted to dynamic situations occur in Motu, where the form is derived directly from a demonstrative meaning 'here (or there) now', and in Chacobo with a progressive translated as 'now'."