I finally had the Lucena position in one of my games

Dafuq? Lucena means building a bridge not burning a bridge.

71. Rd4?

71. Kg7! etc.

Sargon, Kg7 is too early. The black king is a rank too near. After the checks you can't build the bridge cause after rook exchange the black king gets the pawn sinply. Therefore Rd4 is entirely correct (could be winning). Ke8 is the mistake and draws.
Best move is Re7 +. If Kd8, Re8 + and after the rook is protected by the pawn you can play Kg7 and simply move the king towards the black rook to end the checks.

71. Kg7 is also the wrong approach though, since the black king is too close (71 Kg7? Rg1 72. Kf6 Rf1+ 73. Kg6 Rg1+ 74. Kf5 Rf1+ 75. Rf4?? Rxf4 76. Kxf4 Ke7.) One way to win is 71. Rg2 Ke6 72. Rg6+ Kd7 73. Kg8 Ke7 74. Rg7.

I think 71. Rg4 is also quite instructive. e.g. Ke6, Rg6+, Kd7, Kg8, Ke7, Rg7.

Oops, I stand corrected. Move First Think Later… (at least one can return so nothing is spoiled)

Btw, I had it recently as well. lichess.org/siow7pV0

@Harukix I thinkg Ke7 after Kg8 is wrong but starting with 71.Kg4+ is actually interesting: 71.Kg4+ Ke6 72.Rg7 Rf1 73.Kg8 and the pawn is ready to go.