The Catholic Church has a two thousand year history of people receiving under both Species; however, the Roman rite, because of the attacks of some Protestant sects in the 1500’s and in opposition to their statements restricted reception to only the Host in order to make a theological point.
The Western practice of reception under one Species was practiced in certain cases throughout Church history (long-standing customs which give strong evidence, if not proof, against the Eastern Orthodox contention that reception under both Species is Divinely commanded).
It became the standard practice for the West between the 13th and 15th centuries, not the 16th. It was not done as a response to Protestants, since if you start counting Protestantism as a movement with Martin Luther, that’s not until the 16th c. (Luther’s 95 Theses were posted in 1517).
The the Utraquists (from “utraque” which is Latin for “both”, also called the “Calixtines” from “calix” meaning “chalice”), a party of Hussites, rejected the practice of Communion under one Species in 1414, saying that one had to receive
both Species to receive Christ (“utraque” is Latin for “both”). So it was clearly becoming the widespread practice before then, during the 14th c.
St. Thomas (+1274) answers the question, “Whether it is lawful to receive the body of Christ without the blood?” by saying, “It is the custom of many churches for the body of Christ to be given to the communicant without His blood.” (Summa, III, Q.80, A.12) and that’s during the 13th c.
He says that the reason the Chalice is reserved to the celebrant alone in these churches (later made universal in the West), was to guard against profanation from the danger of spilling:
“But on the part of the recipient the greatest reverence and caution are called for, lest anything happen which is unworthy of so great a mystery. Now this could especially happen in receiving the blood, for, if incautiously handled, it might easily be spilt. And because the multitude of the Christian people increased, in which there are old, young, and children, some of whom have not enough discretion to observe due caution in using this sacrament, on that account it is a prudent custom in some churches for the blood not to be offered to the reception of the people, but to be received by the priest alone.” (ibid.)
Anyway, the reservation of the Chalice to the celebrant in the West had nothing to do with “making a theological point” to the Protestants and occurred hundreds of years before Luther was even born. The purpose of this practice is simply to reduce the risk of profanation and in traditional Masses today, the same practice is maintained and for the same reason in addition to it being a long-standing universal practice with regards to the Latin Rite.