"many note-taker's heart" sounds grammatically correct.
Nope, it does not :)
It should, in that case be:
many note-takers' hearts
The construction many a is possibly a bit archaic and certainly a bit poetic, and it does essentially mean the same as many - with the difference that you use the singular in the first, the plural in the second case.
The construction draw extra attention to each individual member of the group that is mentioned.
I worried many long hours for her well-being.
Many a long hour I worried for her well-being.
The first one is quite factual, the second one indicates that every single separate hour counted - it gives much more emphasis on the duration by pointing at the duration of every single hour.
In your sentence, the writer uses this construction to emphasize that for many note-takers, there is a really personal, individual sentiment about pen and paper.