I think we can stipulate that when:
(a) Black has 4 pieces focused on White's Kingside;
(b) White has one piece defending;
(c) White's only other active piece is out of
position;
(d) Black has multiple pins both current and
potential in hand;

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A stock mating pattern that you simply must know. Black mates by force, but you must get the order of the moves right.

1. Ne2+ Kh1 Black starts by forcing the king into the corner. The e2 knight covers both the g1 and g3 squares, meaning that white's king is stalemated.

2. Qxh2+ - not the "tempting" 2. Rh4 because 2...h3 blocks the attack. Note that we could not start with 1. Qxh2+ Kxh2 (forced) because the followup moves are no longer forcing. 2. Ne2 g3! or 2. Rh4+ Kg3! and the mating pattern doesn't work. Black might still win by bringing other pieces into the attack, but why make life harder than it has to be?

2...Kxh2 Rh4#

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After 20.Qxb7? Black played 20...Kxe7 - not to 'win' a bishop so much as to clear h4 for the coming mate; and white duly plays ball by grabbing the rook, 21.Qxa8. Then a pretty mating combo follows.

But Black had a forced mate earlier:
20.Qxb7 Ne2+!
21.Kh1 Ng3+
22.Kg1 Bxf2+
23.Rxf2 Qd1+
24.Rf1 Qxf1#

It may take a move more than Falkbeer's mate, but the forcing sequence starts a move earlier - and aesthetically the result is pleasing too.

In the game line, white could have struggled hopelessly on with something like 20.Qxb7? Kxe7? 21.Nc3 (covering e2), when Black will certainly win but there's no quick mate.

The e2/g3 springboard and the pinned white pawns are a little like the famous <American Beauty> game, S Levitsky vs Marshall, 1912.