How can white force a quick win here? This is from a tactics puzzle I encountered today. The given solution was 1. Qxf6 gxf6 2. Bxf6#,” but to me 1... Ne6 seems to halt white in his tracks, even if it does give up a rook.

My moves as white were clearly not optimal, as it "feels" like there should be a better way for white to bring this game to a checkmate

After 1... Ne6, how should white continue?

[FEN "3rrn1k/pp4pb/2pQ1n1p/q3B3/3P3P/1BP3N1/PP3P2/R3R1K1 w - - 0 1"]

1. Qxf6 gxf6 (1... Ne6 2. Bxe6 Rg8 3. Bxg8 Rxg8 4. Qxh6) 2. Bxf6#
Yes, Ne6 staves off mate, but it does not stave off the loss for long, but more importantly, it does not demonstrate the main point of the tactic, which is why they continue with the "worse" gxf6 in the solution.

Technically speaking, the computer thinks that after 1.Qxf6, that 1...Qc7 is the "best" move, in which case, black should just resign anyway after 2.Qxf8+ Rxf8 3.Bxc7.

There was an error in your analysis that lengthens the game also. 3. Bxg8 allows black to take the queen, and win back some of the material, but does not change the outcome.

P.S. This is a common phenomenon, but sometimes in reverse too. You will see site do this, which I believe is correct since you are trying to teach the main point of a tactic; but I have also seen sites use the computer moves as the main line, and thus, miss the point of the tactic from a human perspective.

  [FEN "3rrn1k/pp4pb/2pQ1n1p/q3B3/3P3P/1BP3N1/PP3P2/R3R1K1 w - - 0 1"]

  1. Qxf6 Ne6 2. Bxe6 Rg8 3. Qf7 {But you are up two pieces, and the attack continues with Nh5 next.} (3. Bxg8 $4 gxf6 4. Bxf6+ Kxg8 5. Bxd8)