I have never heard a priest mention a Mass intention during a Sunday mass
It is done all the time. In some parishes the Mass intentions are published in the bulletin and not mentioned at Mass. In my parish, we have one priest who does not mention the Mass intention, and one that does. But the intentions are always in the bulletin.
Same in my parish.
There is no requirement for or necessity to mention Mass intentions at a Mass. Our parish publishes them in the bulletin, but even that is not required. The parish office usually receives more Mass intentions than can be used at our own parish, and those are sent to the missions. One’s Mass intention is still accomplished, whether or not it is read, published, or even noted! And it may be accomplished by another priest in a foreign land!
It can be, but be aware that when a priest is a Pastor he is required by Canon Law to offer one Sunday Mass for his parishioners. In a parish where there is only one Mass on Sunday, it’s often the case that it’s offered ‘for the parishioners’ and not for any other intention. The same applies to Holy Days of Obligation.
Can. 534 §1. After a pastor has taken possession of his parish, he is obliged to apply a Mass for the people entrusted to him on each Sunday and holy day of obligation in his diocese. If he is legitimately impeded from this celebration, however, he is to apply it on the same days through another or on other days himself.
§2. A pastor who has the care of several parishes is bound to apply only one Mass for the entire people entrusted to him on the days mentioned in §1.
§3. A pastor who has not satisfied the obligation mentioned in §§1 and 2 is to apply as soon as possible as many Masses for the people as he has omitted.