The lack of splitscreen is, sadly, a design tradeoff for having a huge open world where you can drive anywhere.
In most level-based games, like past Burnouts, the whole level is loaded ahead of time. Splitscreen just means having more players in the same amount of space. Every new player in splitscreen comes with a small, fixed overhead cost. The whole level has already been put into memory so there is nothing extra to be loaded.
But in an open-world game like Burnout Paradise, the players could be anywhere. The world is too big to fit into memory, so the game loads as much as possible and then intelligently loads "ahead" of where the player is, so that the world appears to be seemless.
But splitscreen players could be in totally different places on the map, driving at full speed in opposite directions. So the game would have to load twice as much data in the same amount of time. The second player doubles the cost of everything - twice as much memory, twice as much disk bandwidth to load ahead of each player, etc.
There are hardware limitations about how fast textures can be loaded from disk, how much memory is available, etc. Splitscreen is very hard for open-world games. It can be done, but it would take significant resources - making it work would probably tie up their best programmers for months.
Game development is all about allocating your resources as best you can. Ultimately someone decided that it was acceptable to drop split-screen in favor of making sure that the single-player and online experiences were as good as possible, and getting the game out the door on time.
The removal of split-screen still stings, of course :( But maybe you can understand why it's missing.