The meaning and etymology of "cut to the heart"

Solution 1:

The OED has this sense of "cut":

fig. (trans.). To wound deeply the feelings of; to distress greatly. Now chiefly in phr. to cut to the heart.

The first citation they have for this sense of the word is from a bible from 1582, the Douay–Rheims Bible:

When they had heard these things, it cut them to the hart.

This is obviously the second meaning you mentioned ("to be hurt emotionally"). Apparently, it's the older of the two expressions, since "heart" meaning "essential part" only dates back to Shakespeare, judging from the OED's first citation:

Our deere cosin Hamlet Hath lost the very heart of all his sence.
_{Hamlet (1603)}

It's hard to say when exactly "cut to the heart [of X]" came into use. It may be based off the expression "cut to the chase" (which is a film-based idiom that dates back to at least 1929).

Solution 2:

[That] cut to the heart of the larger story I was investigating.

In the sense of the meaning that you are looking for (rather than the Biblical reference), it is important to understand that the phrase is divided:

{[Cut] [to the heart]} {of the subject/matter [in hand]}

In this sense "cut" is, to a degree, a verb of motion implying swiftness of movement:

OED

Cut 18.a. transitive. To pass sharply through, cleave (the air, the water).

a1571 W. Haddon in A. Fleming Panoplie Epist. (1576) 423 Shippes..cut the waves as they are furthered with a merrie winde.

1596 E. Spenser Hymne Heauenly Loue in Fowre Hymnes 69 With nimble wings to cut the skies.

And hence:

19 b. To move sharply, to run rapidly. With various adverbs and prepositions.

1797 B. Hawkins Lett. (1916) 126 He was driving a wagon at the time he was taken, and they cut out and took the horses with him.

1949 ‘M. Innes’ Journeying Boy ii. 25 ‘And now you'd better cut along.’ Captain Cox was a great believer in the moral effects of abrupt dismissals on the young.

Meaning 18a and 19b above create an apt image for "cut to the heart of X" - as it carries both the meaning of slice efficiently with a sharp instrument and implies a swift and decisive action.

To - preposition = as far as

The heart: -

20. With reference to non-material things: the vital, essential, significant, or operative part; the essence or core (of something).

1603 W. Shakespeare Hamlet sig. D3 Our deere cosin Hamlet Hath lost the very heart of all his sence.

1956 R. A. Horn Groups & Constit. ii. 30 Let us cut to the heart of the matter as quickly as we can. We must begin by asking what is the purpose.

1990 Parenting Feb. 38/2 Kids really get to the heart of spelling..when they're engaged in spontaneous writing.

Thus we have a simple combination of words, readily understandable, although mainly figurative, and, as such, a "first mention" is difficult to find as it is not remarkable.

[That] cut to the heart of the larger story I was investigating. = which address directly the essential part of the more complex set of circumstances that I was investigating,