N.B. The email exchange below may have been edited, e.g. to remove content not essential to the main point(s) or to standardize English spelling/grammar.
Do you know whether Tocharian uses the nominative or any oblique case to express the function of the experiencer, e.g. to say "I like something", "I am sad", "I am happy", etc? I [have read] about the development from impersonal to personal constructions in earlier stages of English, where it was still possible to say me thinks instead of I think, and I was wondering how other IE languages handle this situation.
To be honest, I hadn't specifically focused on this question until you asked, so I didn't have a ready answer. I guess the simple answer, in terms of whether the nominative or the oblique (what Tocharianists call the case descended from the IE accusative) is used for the experiencer, is both... or either... or... well, whatever. You can see either of them.
For me, it boils down to the verbal construction: e.g. the me in Lat. me placet is closely tied to the meaning of the verb. So in that sense, many of the verbs that express 'please' or 'be pleased' can be rendered as 'enjoy', and so the experiencer naturally occurs as a nominative. There are, however, a few verbs that take an impersonal construction, for example: Toch. B cänk- 'please' --
I've lifted this example out of Adams' Dictionary of Tocharian B, in case you want to find the source. The syllable me in the above is the enclitic third person pronoun, 'them'... or probably in colloquial English closer to simply "'em", as in "Just give it to 'em!" (Don't confuse it with the Latin 1st person pronoun.) The really nice part about this example is that it in fact comes from PIE *teng- 'think, feel', which gives Lat. tongeō and OE þencan > NE think, so it exactly parallels your original example in English. Pretty neat, eh!
I haven't studied the literature regarding this topic, but I suspect that the shift between me thinks and I think co-occurred with a transfer in meaning (at least the 'dominant' meaning, let's say) from 'seem' to 'think': i.e. me thinks = 'it seems to me', but I think = 'I think, have a cognitive process'. I could be wrong; that's just a guess.
But a similar conjecture might apply within Tocharian. A lower occurrence of impersonal constructions (at least that's my impression, which may not be wholly accurate) may coincide with the rise in Tocharian of causative verbal structures: most verbs have a causative stem, and with many verbs the causative has become so bleached as to be indistinguishable from the base verb, or to have replaced it altogether. The reason I bring this up is that the causative often changes the agent structure a bit (for lack of a better term).
For example, the verb Toch. B swār- 'please' only shows a verbal noun swāralyñe 'pleasure' built to the base form. But the causative B /swāräsḱ-/ has the meaning 'find pleasure in' and shows a conjugated form:
You note here that the conjugation is actually middle: swārästär. In fact the verb only shows middle forms (which makes sense, though we can't make too much of it, since it only shows two forms total!). Anyhow, the point is: with the base form, we'd expect the experiencer to appear in the oblique (accusative); but with the causative, which is what we actually find in finite forms, we expect the experiencer in the nominative. So I think the rise of the causative in Tocharian probably factored heavily in what appears to be a decline of constructions of the me placet sort.