How should this piece analyze his chord progression?

That's not true. From the common sense, this melody starts from F Gong Diao, then quickly turns to C Up-regulation, and keeps C Up-regulation until the end of the second line.

Chord progress depends on what accompaniment it has. I don't know who accompanied the first bar. It sounds strange. And there is no accompaniment behind, so you mean the chord marks of those chords? If so, then those chord marks are the chord progression of this melody.

What you said about C major and B flat major is not reasonable. When you sing this melody, you can feel that it is not a major mode, but a national mode in China.

C chord does not represent c major. Many modes have c chords. In this example, if it is understood as C quotient mode according to what I said, then the C chord used here is actually the main triad of C Ya musician quotient mode.