First, if the AI is unavailable, you at least have a backup plan. You can make changes to your website or web app even if Claude or Codex or whatever else is available. That's useful if you're using third party services.
Secondly, it lets you delve into the code and understand where things have gone wrong if the agent gets stuck in a loop or keeps making poor decisions. If you let the AI handle everything, then you're up a creek without a paddle if said AI gets stuck and can't make any progress.
Finally, it's sometimes quicker and more efficient to make a change without using AI, especially if it's a small scale one. Lots of times I've wanted to add a new form field or tweak some random CSS rule, and being able to do that myself made things a lot quicker.