Timeless Body does not stop your aging

The text on Timeless Body says:

Timeless Body

At 15th level, your ki sustains you so that you suffer none of the frailty of old age, and you can't be aged magically. You can still die of old age, however. In addition, you no longer need food or water.

As a monk, they simply suffer "none of the frailty of old age," which means even as old people, they can still move like they were in their 20's. This makes sense when you consider the amount of old monks who can do flawless kung fu in media, who usually serve as mentors to the younger folk so their techniques can be passed on, and don't die with them.

In conclusion: Timeless Body still lets you age, and you may die of old age naturally. You just don't get the back pain or blurry vision that comes with it, too.