No, one could not just start celebrating the Eucharist or mimicking the Mass and say that they can trace their origins back to Jesus. If I started dressing like you, talking like you, and claimed to be your brother from the same parents does that mean that we are, in fact, related and both come from your parents? Just because someone claims something doesn’t make it true.
The evidence for our being The One Church established by Jesus lies in what we call Apostolic Succession. Our Bishops were all ordained by Bishops who were ordained by Bishops etc. down the line until we get to the Bishops who were ordained by the Apostles who were ordained by Christ himself. Paul tells Timothy: “What you have heard from me before many witnesses entrust to faithful men who will be able to teach others also” ( 2 Tim. 2:2) This shows three generations of Apostolic Succession: Paul, who taught and ordained Timothy, who is to teach and ordain ‘faithful men’ (note that it’s not just any man or everyone, Timothy is supposed to choose specific men to teach).
Another problem with just starting up your own church and saying it was Established by Christ as evidenced by how you mimick the Mass is that Christ’s church was founded on the rock of St. Peter: “You are Peter, and upon this rock I will build my church” (Mathew 16:18). Peter means ‘Rock’ and our Popes are ordained to the same see of Peter. The Church of Christ is founded upon and led by Peter, or Peter’s successors.
The Catholic Church is the only True Church, established by Christ Himself.