The PS2 was basically you buying your first DVD player and also getting a game console with it.
Wasn't the PS3 the cheapest Blu-Ray player available at its launch? Like, it was a free console if you wanted a Blu-Ray player.
The second biggest reason for the success of the ps2 over the Dreamcast was that EA Sports never came to Dreamcast. Sega made the 2k series to compensate and it was definitely a good try, with some 2k titles definitely better in gameplay to their EA Sports competitor. But none of the 2k series ever sold anywhere near EA Sports.