Tire technology over the last 30 years has progressed such that the performance of the tires cannot be easily controlled.
In the early days, the rules controlled rim width. Then the rules needed to address the tire width. Then the rules were written to extend tire mileage to slow down vehicles. The manufacturers always found way around the rules by developing new tires and compounds. In my opinion, the harbinger of the end of true tire wars was Michelin's Optimised Contact Patch, which was a holistic concept that connected vehicle suspension design, with tire carcass suspension, and tire compounds. The result was a significant increase in the contact patch, particularly during cornering. This technology propelled Renault and Fernando Alonso to consecutive F1 titles in 2005 and 2006. Bridgestone developed their own holistic approach to tire design, as well.
In 2007 the MotoGP rules changed substantially, and performance was more or less governed by fuel efficiency rather than capacity. This put pressure on the tire manufacturers to increase the lateral acceleration of the bikes, particularly during corner entry, since less fuel was available on corner exit. This performance paradiagm was deemed unsafe. In my opinion, a deal was brokered behind the scenes to keep the tire war alive in 2008, but Bridgestone decided to bring some F1 tire technology into the MotoGP fold. Michelin was not interested in following suit, particularly without Rossi, and they withdrew from MotoGP. The 2009 season was not a control tire, rather a single tire supplier, but it's a distinction without much of a difference.. The 2009 season was effectively the end of true tire wars for any form of racing on paved circuits.
The "tire wars" that exist now are mostly technological kabuki. It's more about price point and serving customers, and building tires to fit a certain performance niche that generates publicity for the company. The technological pathways to dominating a sport are limited by the regulations, and the tire manufacturers want it this way. This kind of tire war currently exists in FIM EWC (endurance racing).
World Superbike also had a kabuki tire war arrangement throughout the 80s and 90s, but it was abandoned for different reasons than MotoGP. When grand prix racing moved to 990cc four-strokes (2002), the Flamini Group, who were in charge of World Superbike, wanted to increase tire performance. Pirelli had no presence in the grand prix paddock so they were happy to work with the Flaminis. World Superbike introduced a Pirelli control tire in 2003 to raise the cornering performance of superbikes, and make lap times more similar to MotoGP.
