Why can't non-Catholics receive Communion at Mass?

It is not a judgment on anyone's holiness. It is about what Communion means.

Communion is a statement. To receive the Eucharist is to say "Amen" to everything the Catholic Church believes: about the Real Presence, the priesthood, the sacraments, the pope, all of it. St. Paul says that "because the loaf of bread is one, we, though many, are one body" (1 Cor 10:17). Receiving while not sharing that faith would make the sign say something untrue (CCC 1398-1401).

Catholics are held to the same standard. A Catholic in mortal sin, or one who rejects Church teaching, is also asked not to receive (CCC 1385; Canon 916). The rule is not "Catholics in, others out." It is "those in full communion with the Church, in a state of grace."

Exceptions exist. Eastern Orthodox Christians may receive in a Catholic church if they ask and are properly disposed, since their sacraments and Eucharistic faith are the same (Canon 844 §3). Other Christians may receive in danger of death or grave necessity if they share Catholic Eucharistic faith and cannot reach their own minister (Canon 844 §4).

What visitors can do. Come forward with arms crossed over your chest for a blessing, or stay in the pew and pray. Many converts say the years of not receiving taught them how much the Eucharist matters. And if you want to receive, the door is open: talk to a priest about becoming Catholic.