I think Whitman carefully uses the word SHOULD. Americans in particular have a very resilient, hopeful, can-do, pessemistic, hard-working attitude.
The words blithe and strong fit this model of an American. The fact that he uses the word SHOULD points out that we don't always fit this and that not all people do fit it. Look at the eras of American history that Whitman lived through and that followed him. We have been through pain, depression, failure among other problems. Our peoples have experienced that too. Whitman clarifies that we are a work in progress by using the word SHOULD.
The speaker says this because that is how he thinks the spirits of average Americans should be. This is what he thinks the American soul is like.
In this poem, the speaker is talking about his vision of America. It is a vision in which all sorts of regular (not elite) Americans are showing their spirits -- showing what kind of people they are.
Blithe means something like happy and carefree, lighthearted. To the speaker, the typical American is like this. The typical American feels free and lighthearted but is, at the same time, strong.