A haphazard (no to say random) check of the uses of 'pilin' in the corpus at tokipona.net yields these results. Far and away most uses are as a verb (immediately after 'li'). The noun uses are mainly as objects of 'jo' and are mainly of the 'pilin pona' form, though several 'pilin ni:' cases also occur. In the verb position, over half the cases are of the 'pilin pona' type (though more 'ike's than 'pona's, with a few 'nasa's and 'sama's -- from so long ago that it still took 'e'). Almost all of the remainder are 'pilin e ni:', with an occasional 'pilin e kon' "smell" and 'pilin e uta kepeken uta' "kiss" as well as other body parts and the odd physical object like 'kiwen' or 'telo'.
Based on this, it seems clear that by 2009, when the corpus was compiled, 'pilin' operated in at least three ways (from an English point of view, at least, but clearly distinct even in tp). The least significant of these is, roughly, "touch", getting sensation by contact or even just the contact alone. This is tructurally similar to the much larger class, roughly "think that, believe", which take, in effect, a propositional object, indicated by 'ni:'. The sample turned up no clear cases of 'pilin' for thinking in a process sense, unless the two specimens of 'pilin pi x' are cases of "thinking about", as was possible early on. At least possbly related to this are some cases of the noun 'pilin pona', which could be (in the limited context given) good ideas as well as good feelings (in either sense). But for the most part, 'pilin' here matches English "feel" pretty closely and touches only accidentally on English "think".
So, 'pilin pona' matches ''feel good" perfectly, right up to (it turns out) the difficulty in figuring out how it works grammatically. In English, the obvious answer is that "good" is the object of "feel", but the story of of the man who had to fish an oject out of the back of a pretty woman's jeans and in embarassment remarked "I feel a perfect ass" shows this is probably wrong for the case we are considering. The second possibility is that "good" is an adverb modifying "feel". The fact that "good" is usually an adjective with "well" as the corresponding adverb does not discount this suggestion, since "well" can be used in this same construction in exactly the same way -- as well as in the "normal" way of evaluation performance at feeling (odd as that may seem). So, if it is an adverb, it is a different kind from the usual evaluative one, but what kind? At this point we need to consider the differences that the subject makes: "I feel good" is pretty straightforward, but "He feels good" goes in at least two directions (skipping the evaluative): a report about that he would say "I feel good" and a report of the speaker's response to contact with him. In short a to-him/ to me contrast. The latter is clearer with a different term: "This stone feels cold" clearly does not mean that the stone has feelings; rather I have feelings about the stone: that it is cold (whether it is or not is beside the point). And this brings us back to "I feel good", since again whether I am good or not (in the appropriate sense) is not strictly relevant. That is, "I feel good" amounts to "I feel that I am good". The grammar of the collapse is messy, though not unprecedented. The motivation is the usual one of caution. If I say"I am good" (in the understood sense) then I can be easily proven wrong by things like crying, going into a rage and so on. But If I only feel that I am OK, then I am at worst misguided.
This excursion into English need not have a direct effect on the case in tp. However, it does suggest some issues. Taking 'pona' as an adverb to 'pilin' has the usual result of giving an evaluation of performance, which is clearly not intended. Taking it as a direct object also doesn't seem to fit as there is no external object to be felt. (The grammar of this move is carefully not discussed in the textbooks, which are trying to do the rudiments. Once those are mastered, the other tricks, like incorporation -- assumed here -- can be deployed carefully. After all, at some point we do get 'jan pi alasa waso' from 'jan li alasa e waso'.) . As in English, the move from 'mi pilin e ni: mi pona' to 'mi pilin pona' would be complex, but does seem to provide an explanation for the behavior of the expression and a place to put some of the occasionally needed but missing information. It also unifies the various fields of 'pilin' (except the bit about touch)
One thing that this show that is hidden in both the English and the tp is that the good/pona is ultimately a property of the subject, not of the feeling. This appears in the English in a reanalysis "I am feeling-good" (not an impossible -- though an implausible -- reading). This could derive (rather more simply, in fact) from the same underlying sentences as the above reading. And the tp version of that (under the usual rules) would be 'pona pilin'. Indeed, 'pilin pona' as a translation of "feel good" was always odd unless 'pona' was an object, given the shift from AN to NA. Which looks like the 'pilin pona' form is actually an unthinking calque of the English and, so, to be avoided.
The obvious problem with all this is that, under this analysis, 'pilin pona' has nothing to do with my emotions or the like, but only with my beliefs. It does not refer to an internal non-cognitive state, but only to a cognitive one. It is only a cautious claim about some objective truth, not a strong assertion of a subjective one. And that may be unfair to what 'pilin pona' actually does: I may feel good despite all the contrary evidence, even as known to me. But then again, this is a special kind of good, then, as 'pona pilin' and 'feeling-good' suggest.
[This has been an exercise is Humean reasoing. I intuited my conclusion and worked overtime to build a case for it. Now to face the court of common sense.]