Nobody in this thread has posted a single argument supported by data. Here's the only data you need to know.
Pitcher aging curves:
http://www.fangraphs.com/blogs/pitcher-aging-curves-introduction/
Slope of the FIP line increases after 26, and again after 30. Velocity decreases after 26, then tanks as they approach 30.
Another:
https://www.beyondtheboxscore.com/2014/2/25/5437902/pitching-aging-curves
Again, pitcher value peaks at 26, and is roughly 75% of peak by age 30. The number of pitchers in the league follows the same curve, meaning they are out of baseball either through injury or poor performance.
So yeah, all available evidence suggests it's wise to dump pitchers after their peak at age 26.
It doesn't matter what you think, or what you want to see your team do. Pitchers peak around 26 and then decline rapidly. The best way to maximize overall value from them is to dump them before they decline too much. Meaning, get rid of them before they hit 30...
which is why my process trades them by age 28-29.