In a nutshell, if my quadripartite system distinguishes four sets of endings exemplified in the 1ps with *-mi (objective progressive), *-m (objective non-progressive), *-h₂ór (subjective progressive) and *-h₂e (subjective non-progressive), then it stands to follow that there may likewise be four non-finite forms, participles, corresponding to each of the four categories I describe. From the evident participle suffixes, we then seem to be inevitably led to the following system:
However, Szemerényi informs us in Introduction to Indo-European linguistics (1996) that the PIE *tó-participle is just not present in Tocharian and Anatolian. From this absence of evidence, it's understandably concluded that the participle hadn't yet formed. Only once Anatolian and Tocharian parted ways would the emerging Core IE dialects create this new participial form.
This status quo account is admittedly very persuasive... as long as one forgets to question how such a suffix can be formed from known PIE grammar specifically with the required semantics to make it the prevailing participle form by far, above all other possible thematic suffixes like *-nó-, *-mó- and *-ló- among others which are also occasionally used. Why did all of the Core IE dialects agree to this one suffix with *-t-? I have a defiant answer: Maybe it had been a participle ending right from the start and that the Anatolian-Tocharian area were motivated to chuck this one ending away. But then, why?
Falling back on my recent insights on how Anatolian-Tocharian emerged out of my model (note too my later relabelling of "eventive" as "progressive" in this model), I realize one interesting motivation for a conjectural loss of this participle. Notice that my theory suggests that Anatolian-Tocharian dialects were developing tense out of a tenseless system, making the former progressive marker *-i a present tense marker. In effect, the four-way system of old was reshaped into a three-way system of mi-class, hi-class and middle. That means that one of the participles had to go, and guess which one! So these dialects must have ended up with a mi-class participle in *-ónt-, a hi-class participle in *-wós- and a middle participle in *-m(h₁)nó-. There would have been no longer any room for a *tó-participle in this particular evolution as it would only duplicate the function of one of the other three, hence a loss specifically in Anatolian and Tocharian of a now-redundant element.
 Szemerényi, Introduction to Indo-European linguistics (1996), p.323 (see link): "The suffix -to- is widespread in all IE languages except Anatolian and Tocharian." Well that was pretty straight-forward, wasn't it?