What is lost in the passage from sati to mindfulness is worth naming precisely, because both the enthusiasts and the critics tend to overstate their case.
Removed: the doctrinal fourth field of observation, the goal of liberation from rebirth, the ethical precepts as a precondition rather than an optional extra, the monastic setting, the sustained teacher-student relationship, the surrounding community, and the analysis of experience as empty of self. That is not a trivial list, and calling the remainder Buddhist practice is inaccurate.
Retained: the technique of returning attention to a chosen object, the stance of non-reactive observation, the body scan, and the framing of thoughts as passing events rather than as reports about the world. These are real, they are not nothing, and they are the components to which most of the measured effect is attributed.
The critique from Buddhist teachers is that a practice designed to dismantle attachment has been repurposed to make people more comfortable inside the arrangements producing their distress, and the coinage McMindfulness has stuck to it. That is a serious argument and it is not answered by pointing at outcome data. The counter-argument from clinicians is equally serious: they are treating illness, they say so plainly, and they are not obliged to hand a soteriology to people who came in with chronic pain.
The one claim that should not be made in either direction is that clinical mindfulness is simply what Buddhists have always done. It is a derived technology, honestly derived and openly documented, and the tradition it was derived from is still there and still doing something else.