The chalice had stopped being offered to the laity as a matter of custom at some point before the early 15th century, as this is when it began to be a significant theological issue in Bohemia, with Jacob of Mies proposign in 1414 that communion must be offered "under both kinds" (the common English description of taking both bread and wine) to the congregation, leading to a movement called Utraquism, part of the Hussite reform movement. It continued to be an issue through the reformation: the practice of reserving the chalice for the celebrant is explicitly condemned in the XXXIX Articles in England under Elizabeth I in 1571 ("XXX. Of both kinds. The Cup of the Lord is not to be denied to the Lay- people: for both the parts of the Lord's Sacrament, by Christ's ordinance and commandment, ought to be ministered to all Christian men alike."). Defense of the practice was part of the Council of Trent, in particular Session XXI from the year 1562.

Much of the ink spilled in Western Europe on the subject is either Reformist/Protestant insistence that it must be offered in both kinds, or Catholic defense that only the bread is necessary. The reasoning for why it should be withheld to the priest is logistical at its root -- it is much easier to spill the wine and thus gravely disrespect the Real Presence of Christ in the Eucharist than to make such an accident with the bread -- see here for an example of this argument from a traditionalist (and thus not mainstream RC) church. Reformers argued that there was also an element of sacerdotalism in the practice -- a trend through the Middle Ages had been to emphasize the separation between laity and clergy, and reserving the chalice was an outward way of doing so.