Hi Paul, good post - plenty to work through there.
I have also checked out the validity of my line from my post above, but first, for the sake of completeness I'll show the mate if Kxd5.
[FEN "r1b3n1/ppp2qb1/2kp4/5PB1/3P2PQ/2PB4/PP3K2/4R3 w - - 0 1"]
1. d5 Kxd5 {if Qxd5 Be4 pins and wins the Q} 2. Qh1 Kc5 3. Be3 Bd4 4. Bxd4 {is a nice mate with all the diagonals covered}
Now to have a further look at Kd7 Re6 with the idea of f6 and Bf5
[FEN "r1b3n1/ppp2qb1/2kp4/5PB1/3P2PQ/2PB4/PP3K2/4R3 w - - 0 1"]
1. d5 Kd7 2. Re6 a6 {a null move to check whether white is actually threatening anything} 3. f6 Bf8 4. Bf5 Kd8 {There are other moves but this walks out of the discovered check and appears to threaten B takes rook} 5. Qh8 {engine top choice, keeping all the escape squares on h file covered and adding pressure to the g8 knight. Now if...} Bxe6 6. dxe6 {takes away most of the black Queen's squares} Qe8 7. f7 {A winning discovered check with one of several forced mates to follow }
So onto the line in Pauls analysis (Kd7 Re6 Bf8)
[FEN "r1b3n1/ppp2qb1/2kp4/5PB1/3P2PQ/2PB4/PP3K2/4R3 w - - 0 1"]
1. d5 Kd7 2. Re6 Bf8 3. f6 Kd8 {same ideas as above but a move earlier} 4. Bg6 {a nice move which if taken...} Qxg6 5. f7+ Be7 {one of several moves to block the discovered check - all fail} 6. Rxg6 {leaving White a Queen up for a piece or two and forced mate in 6 to 8 moves}
This analysis is not exhaustive and I hope not exhausting as I have stuck to lines I understand rather the more esoteric lines the engine tries to in effort to wriggle out of loss.
I often wonder how much a player like Steinitz would have seen at the time and how much he might leave to chance on seeing that he had good lines after d5+. It also strikes me that to see much more than the mate after Kxd5 and Steinitz's piece win that is the solution, in the 40 secs the video in the OP gives is asking a bit much.