tl;dr: There is some basis for the Christian interpretation in Jewish sources, but it is not the only reason for the prohibition.
In the Bavli it is stated:
ת"ר החוסם את הפרה ודש בה לוקה ומשלם ד' קבין לפרה וג' קבין לחמור
(Bavli Bava Metzia 91a)
Our Rabbis taught: If one muzzles a beast and threshes therewith, he is flagellated, and pays [to the owner of the cow] four kabs in the case of a cow, and three kabs for an ass. (Bavli Bava Metzia 91a, Soncino Translation)
When one muzzles a cow or donkey while threshing, one must pay the owner of the animal for the food the animal would have eaten if it was not muzzled. However, in addition to the payment, one also receives lashes, indicating that there is an additional aspect to the commandment, besides compensating the owners.
In addition, one is prohibited to muzzle the threshing animal, whether one loaned / rented the animal or one owns it.
Thus, while there seems to be some basis for the Christian interpretation, there must be an additional reason for the prohibition:
משרשי המצוה. ללמד עצמנו להיות נפשנו נפש יפה בוחרת הישר ומדבקת בו ורודפת אחר החסד והחמלה, ובהרגילנו אותה על זה אף על הבהמות שלא נבראו רק לשמשנו, לחוס עליהן לחלק להן חלק מיגיעת בשרן תקח לה הנפש דרכה בהרגל זה, להטיב אל בני אדם ולשמר אותם מהעביר עליהם הדרך בשום דבר שראוי להם, ולשלם שכרם ככל אשר יעשו טוב, ולהשביעם מאשר יגעו בו, וזה הדרך ראוי ילכו בה עם הקדש הנבחר.
(Sefer HaChinuch, Mitzvah 596)
Among the roots of the Mitzvah: To teach ourselves to make our soul a beautiful soul, which chooses right and keeps to it and runs after kindness and compassion, and when one makes it [the soul] used to this even with animals which were only created to serve us, to spare them to give them part of their body's work, then the soul would get used to this, and learn to be good to humans and guard from taking from them anything they deserve, and to pay their salaries when they do well, and to feed them from what they worked in, and this is the path which the holy chosen nation should walk.
(Sefer HaChinuch, Mitzvah 596, my translation)
According to Sefer HaChinuch, muzzling the animal while it is threshing is cruel, and the commandment teaches us to avoid cruelty, even to animals.