I must say here I did not count at all: I simply felt.
Like you, I'm not a big believer in metronome use, but sometimes it really is necessary to either use one or to rigorously count out the music at the shortest unit contained within it, in this case at the semiquaver (16th note) level.
I do understand and certainly appreciate this, Richard. As Andrew mentioned, and I'd bet most players agree, the metronome is also not something I enjoy (to put it gently).
Personally, and this may branch from my lack of formal training in piano pedagogy (except on the receiving end), I'm quite diffident about specific preparation practices. I'm sure this metronome topic has been beaten to death in numerous threads on this site, and likely by pianists/teachers much more knowledgeable than I.
For me, I use the metronome as a tool only when I need it - applied to specific sorts of passages to correct undetected (and inappropriate) tempo changes. I also, and others may differ here, never use it early in the learning "life cycle". For me I will have progressed on the piece considerably, learned all the notes well with some definite muscle memory kicking-in, and am beginning to focus more on musical aspects before I consider introducing a metronome for areas I feel need it. Typically I use the metronome in short spurts, as required.
I don't think you were implying that everyone who utilizes a metronome as a tool doesn't play with their "heart" or with "feeling". I do think perhaps sometimes this is true, and it is quite easy to detect IMO in the final result. I set the metronome aside well before any performance or recording, so as not to interfere with the real goal - communicating the music itself!
Whatever the method - listening back to recordings, tick-tocking in your head, taping your toes, or a metronome - I do firmly believe applying some rhythmic discipline is a necessary step on the path to good playing.