Am I getting this right?

1) bilirubin (from red cell breakdown) from bloodstream excreted into the gut

2) gut biota convert it to urobilinogen via bilirubin reductase

3) urobilinogen -> urobilin spontaneous breakdown (still in gut)

4) urobilin reabsorbed into bloodstream, filtered by kidneys, excreted in urine

If so, why not just leave it in gut for excretion with feces?

or perhaps 2.1) urobilinogen reabsorbed into bloodstream first, then 3). Makes a little more sense