Is he actually trying to claim that anyone outside of the Catholic Church is barred from receiving the Eucharist?
This is the practice of the Catholic Church, not just this one individual, and for the very reasons cited above. This is also true in the Orthodox Church. In Church missalettes, it reads:
For our fellow ChristiansWe welcome our fellow Christians to this celebration of the Eucharist as our brothers and sisters. We ray that our common baptism and the action of the Holy Spirit in this Eucharist will draw us closer to one another and begin to dispel the sad divisions which separate us. We pray that these will lessen and finally disappear, in keeping with Christ's prayer for us "that they may all be one" (Jn 17:21).
Because Catholics believe that the celebration of the Eucharist is a sign of the reality of the oneness of faith, life, and worship, members of those churches with whom we are not yet fully united are ordinarily not admitted to Holy Communion. Eucharistic sharing in exceptional circumstances by other Christians requires permission according to the directives of the diocesan bishop and the provisions of canon law (canon 844, paragraph 4).
I take it you were not aware of this. Think about it ... the word "Communion" is what unites same believers.