Should I hand over the door that serves as the anchor to the Thief of Time?
This is actually a simple choice, because the options do not involve changing a major variable in your life.
It's like a classic scenario where your loved one and your blood relatives fall into a river at the same time, whom would you choose to save? Stripped of other conditions and with only two choices, no matter whom you choose, it will bring a great variable to your future life, because it's difficult to have the best of both worlds.
However, the choice given by the Thief of Time this time is not like that.
If you hand over the anchor, there might be some variables to a certain extent; but if you don't, it essentially equates to no change.
When you have confirmed that the Thief of Time has impure motives, then the best way to defend is to try not to change the status quo as much as possible.
Therefore, Angel's choice can't be said to be absolutely correct, but it will definitely not involve risk.