In many shuls, I see the majority of people begin putting away their tefillin when the *chazan* begins the full Kaddish (after *U'va Letzion*). To me this seems disruptive and disrespectful to the mourners saying *Kaddish* at the end. (Just because people don't complain, doesn't mean that it may not bother them.)

1. Is that disruption/disrespect an actual concern cited in authoritative sources as leading to a recommendation/requirement not to remove the tefillin at that time?
2. Are the tefillin supposed to be worn for the entire duration of Shacharit (except for Rosh Hodesh / Hol Hamo'ed and those who switch to Rav Tam, etc.)? Doesn't Shacharit end after the last Kaddish? (Nusach Ashkenaz - I think Sefarad is the same idea.) Is it proper then that people put away their tefillin before the end of davening?