You ears are likely deceiving you. The traditional explanation of the pronunciation rules, which widely appears in textbooks and Japanese pronunciation manuals, is absolutely correct, and also in line with what is normal for human language (e.g. how do you pronounce "input" in normal speech?).
I mean, it's conceivable that this was a pedagogical technique or a strange dialect of my teachers, but when they modeled pronunciation, it was clearly with an "n" sound. Or perhaps when they spoke each mora distinctly and somewhat separately, they unconsciously reverted to the "normal" pronunciation of ん instead of a modified pronunciation.
Now, maybe having modeled that, when they spoke at normal speed it was an "m" sound and, not listening for it, I never heard it.
(Also: They never corrected my pronunciation of ん, which was consciously an "n" sound. They did correct my pronunciation of other things, like failing to subvocalize vowels between unvoiced consonants (I remember a long and arduous exchange in which my professor pronounced "hito" (人), and I tried to mirror it back, imagining that my problem was in not emphasizing the second syllable enough rather than failing to subvocalize the "i"). And I think that I say "input" with an "n" sound. But maybe I'm incapable of saying "input" the way I normally do in speech when I'm actually thinking about it. :P )
But if you still doubt, I grabbed the first three audio recordings of "tempura" I could find in videos. All are very clearly bilabial nasals (i.e. "m"): https://www.youtube.com/watch?v=4b6glqZKVBY https://www.youtube.com/watch?v=RUslqCQd4W4 https://youtu.be/56PHtLWJh-0?t=159