As a different view, I would say that this is "wrong" in the sense that we usually expect transitivity with (many of) our relations. E.g. if I write $a=b=c$ then usually we would say $a=c$ as well.
There are certainly counterexamples for more general relations, such as if $aRb$ means $a$ and $b$ share a hobby, so that $aRbRc$ wouldn't necessarily mean $a$ and $c$ share a hobby (maybe $b$ just has lots of hobbies).
But we don't use notation that has an unambiguous interpretation as an ordering for that. And $<$ definitely has that interpretation, so unless you want $$x<a<b<x<a<b<x\cdots$$ then the notation you suggest is too open to misinterpretation.
Interesting side note: this ambiguity is actually relied on in notation for outcomes of voting methods in the literature. In that case, $A\succ B\succ C\succ A$ has meaning when $A\succ B$ means $A$ defeats $B$ in a head-to-head vote - because we don't always get transitive election results. So sometimes our notation has to be careful depending upon who is reading it.