The odds of a tie rod breaking are near zero but the odds of a tie rod END breaking are much higher.
There’s not any misinterpetation over what is needed is there? Some of the auto parts websites often have these parts listed incorrectly. (AutoZone is one of them)
Tie rods are inners attached to the steering rack. Tie rod ends connect the tie rod to the steering knuckle.
No matter which is needed the price quoted is fair and assuming the tie rod diagnosis is correct and the tie rod ends are tight there is something else that should be considered.
Tie rod ends may be tight and still be faulty. Sometimes the only way to determine this is by separating them from the steering knuckle and this will occur when the tie rods are changed. Once disconnected the ball studs in the tie rod ends should be rotated by hand to make sure there are no flat spots in the ball studs. If there are these should be replaced along with the tie rods.
Inner tie rods can be tightened but this is not something that should be done by a professional shop for a customer as it would be considered an improper repair. For a DIYer it can work fine. The tie rod is removed from the steering rack and put into a press and recrimped with a heavy metal sleeve.
Don’t expect a shop to do this though.