Unless my reading has misled me the act of canonisation is now seen as an exercise of infallibility by the Pope. So it seems eminently sensible that he should be doing the proclaiming for all his Church.
However beatification is not such an eminent thing and is left to promulgation by a cardinal appointed by the Pope.
Venerability is left to the local bishop.
Beatification, such as it is, is left to a consistory acting in the name of the pope; its cardinal head making the announcements, when the pope doesn't do so.
Sainthood itself is determined (normally) by this same consistory. But as an infallible declaration of someone being in heaven, it's final check is the pope accepting and promulgating the decision of the consistory.
As to why easterns take longer? The two miracle requirement. Few Romans pay attention to the Beattifications of Easterns.
Martyrs usually hit beatification right quick... but then the wait for the documentable posthumous miracles begins.