Here's a different version of the problem.
It takes 10 minutes to walk home from the bus central. The bus is late but should be here any minute now. The bus takes one minute. Do you wait or walk?
It takes 10 minutes to walk home from the bus central. The bus is late but should be here any minute now. The bus takes one minute. Do you wait or walk?
You want to optimize for when you get home, not for your health or environment.
If the question was 1h+ then maybe the answer would be different.
There is a strategy that allows you to never be worse than 2x if you knew exactly when the bus arrived: Wait for 10 minutes, and if the bus didn't arrive, walk home.
In all cases when the bus arrives between now and in 10 minutes, you do the optimal thing, and whenever the bus arrives after 10 minutes have passed, you will be home after 20 minutes, which is not worse than 2x worse than optimal.