Why is urobilinogen reabsorbed from the gut just to be excreted via the kidneys?
I think that it is missing some details. Note that the vast majority of urobilinogen (~80%) is actually eliminated via fecal elimination. I would recommend looking at the article on enterohepatic circulation, which provides more context on this issue. Basically, bile salts can be useful (even or especially after metabolism by bacteria), so a fraction is reabsorbed by the large intestine and then dumped back into the digestive system:

However, just like bile, some of the urobilinogen reabsorbed is resecreted in the bile which is also part of enterohepatic circulation. The rest of the reabsorbed urobilinogen is excreted in the urine where it is converted to an oxidized form, urobilin, which gives urine its characteristic yellow color.


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The net effect of enterohepatic recirculation is that each bile salt molecule is reused about 20 times, often multiple times during a single digestive phase.

Obviously urobilinogen isn't one of the salts recirculated that much if most of it gets excreted in feces, but apparently it still happens (possibly just because it is chemically similar to the other bile salts, as you suggest).
I think that the mechanism which is emphasized in the urobilinogen wiki page is a secondary mechanism for excretion when urobilinogen is at toxic levels in the bloodstream (e.g. jaundice). These high levels come for example from blood breakdown as pointed out in the article, which may be emphasized because it is a common medical issue (which is of course not a focus of SE Biology).
So, in other words, the wiki article is targeted towards the potential medical problems (toxic levels requiring elimination via kidney) rather than the normal physiology (mostly eliminated in feces, sometimes-useful salt that helps digest stuff), possibly leading to confusion. This flash card resource succinctly notes the functions of bile salts such as bilirubin and its derivatives and may be helpful for other basic info:

Functions:
(1) Digestion and absorption of lipids and fat-soluble vitamins
(2) Cholesterol excretion (body's only means of eliminating cholesterol)
(3) Antimicrobial activity (via membrane disruption)

More about (unfortunately medical focused) physiology of bilirubin etc. can be found here.