tags modify a complete selbri, not a single tanru unit, so PU (or any other tag) makes CU unnecessary.

A tense tag like "pu" will split tanru. (Which is handy for making sentences flow well!) No, it doesn't only affect a single brivla, only a whole selbri. "na'e" works the way you're saying, for instance you can say "lo na'e broda brode" or "lo broda na'e brode" and either way you get a tanru with a different element negated. With "pu" you can only tag a sumti (saying that the bridi event happened before the event described by the sumti) or you can attach it to the front of the whole selbri (saying that the bridi happened before some unspecified event, which is our idea of past tense).

In the circumstances where you might have wanted to use "pu" in a tanru, I recommend the gismu "purci". Or here's another interesting shape: "lo cizra nu pu klama cu fasnu", A strange event of going occured. "pu klama", went, which is a tagged selbri, is further tagged with "nu" (which tags a full bridi), so we have a nu/[kei] element that describes an event, and then THAT can take the seltau "cizra", strange. But that's a little strange. :)