That's all news to me - I have literally played Taps at hundreds of funerals and that's not how it went, mainly because the firing party were also the flag folders, so the flag couldn't be presented before the volleys - it had to be done afterward. In my sequence it was:
Firing party does "stack arms"
Firing party marches up to the casket and folds the flag
Flag is presented to the NCOIC in charge of the honors detail
NCOIC presents the flag to the next of kin and salutes
That's the basic gist of it - it was a 7 person detail with 5 members of the firing party, plus FPC and the NCOIC.