Man, I remember all this stuff. PS2 games would have ADX and RenderWare logos on boot. Good times.
Even personal favorites like Grand Theft Auto III and Persona 3FES used RenderWare. After the 6th gen, Unreal and Unity became popular amid rising development costs. During that time, I was surprised when companies insisted on making their own engines.
An example of this is Square-Enix's Luminous Engine. While Kojima Productions' Fox Engine was used in twelve games, only three games used the Luminous Engine. (I just did the research. I thought the Fox Engine was only used for Metal Gear Solid V and Metal Gear Zombies Survive.)
I don't have the knowledge to talk about optimization or efficiency compared to modern alternatives, but I will say that the games that used RenderWare seemed to benefit highly from it. The PS2 in particular was known for its complex architecture, so intermediaries were a good choice to speed up and streamline development.