You are right that the French Open is the only Grand Slam tournament without Hawkeye,
The reason is that Clay, unlike grass and hardcourts, leaves a mark that players and umpires can check and verify if the ball was in or out.
You can argue that some mistakes can be made but that's quite rare, A hawkeye system is very expensive and the cost/benefit ratio doesn't pays.
Gilbert Ysern, the director general of the French Open and a former umpire, address to this:
“I don't think we need it,” he said in an interview in his office.
“There are ball marks on clay,” Ysern said with a genial smile, “and
our chair umpires are used to checking the marks when needed, and, so
why would we need Hawk-Eye?”
And the disputes? “It happens very, very rarely that the officials
can't find the mark,” he said.