If you're actually burned out, you'll be staring at it for the whole 25 minutes, then getting extra stressed out at reporting it to the accountability partner, which results in a downwards spiral. Heck, you'd be lucky to even finish writing the plan.
Overcoming burnout is looking at your situation from the outside in. What you described is clearly a downward spiral. Logically, wouldn’t it be better to write some words for a plan than no words? If you can procrastinate for 25 minutes while you are burned out, you can spend 25 minutes writing a plan, even if it’s one word a minute or one word every 5 minutes.
The key is to break down what you need to accomplish and how you will accomplish it in the smallest incremental steps possible. Once you have that list, it becomes less overwhelming to make progress. Don’t focus on completion, focus on making tiny incremental progress.
I think internet health advice can be harmful because not everyone has an equal definition of the same terms. Society tends to use the same terms for fatigue, burnout, and plain laziness. It's like trying to apply aloe vera to a third degree burn.
I think you're on to something with trying to regain control in small incremental steps. But it's probably another approach.