So, now that we have established, the asylee can get travel document, why does he/she need a passport?

People who say obtaining GC by any other means is similar to getting one via asylum route seem to have confused with the motive. Asylum seeker have declared that the country that they were citizen of, is not safe for them. Other GC holders are thus not under the same category. Why is this important? Because you are trying to get travel document from the same country that you said isn't safe for your habitat.

Now if you think the hostile situation no longer exists and you should be allowed back in the country then you have to surrender you asylum status.