I’m not the most knowledgeable person, but honestly I think of refusing blessings to the sick, the young, the elderly, etc as a little like denying bread to a starving man. I believe there can only be good that comes from knowing someone has prayed for you, especially from the consecrated hands of a priest. In a manner of speaking, the same hands that physically hold Our Lord are now holding you. The same voice that speaks the sacred words of the consecration is now speaking a prayer for you.The priest, who we say is a representative of Christ on Earth is also, in that moment, a representative of Christ’s deep personal love for you. It’s really a beautiful thing. I feel sad that those who are sick from physical ailments are being denied that spiritual comfort.
Perhaps it would best avoid confusion and the resulting gossip (which truly can destroy congregations) if this particular priest would simply explain the reasoning publicly and clearly? Perhaps in homily, if appropriate? I didn’t know about blessings not being recommended during mass itself until reading this thread so it’s likely the others in the congregation don’t know the reasoning either. I leave the question of blessings during mass up to the judgement of the Church.
But outside the mass I believe blessings have been given for centuries. I don’t know of a good reason to deny them to someone who is asking because they believe they are in need of one. These are simply my thoughts, so if I am missing something, then feel free to let me know.