The old LMP demo format only contains the player's actions: e.g., whether the forward key is pressed down, or whether the Fire is pressed. The old demos assume that game events always occur exactly in the same way, if the input is exactly the same. In other words, there is no real randomness.
Unfortunately, Doomsday has been changed so radically when it comes to the internals of the engine's network code and jDoom gameplay code, it is impossible to remain 100% backwards compatible with the old gameplay logic. If the user input of an old LMP demo is played back in the current jDoom, the game events will very quickly diverge from those that occured when the demo was recorded.
The only way to get around this would be to write a new demo recorder and embed it in the original source code. However, this means a great deal of work: Doomsday demos contain the network packets sent between the server and the client. The old source code would have to be augmented with the entire server side network packet generation subsystem.
Since Doomsday 1.0, the original demo format has not been supported. Revising the demo format made the demos much more powerful: they can now begin and end at any time during a map, for example.
You can still use the old jDoom version 0.99.6 to play the original Doom demos, albeit with less eye candy.