Today, as in times past, courts of law hold that what a man does through a duly constituted agency, he himself actually and legally does.
Personally, I think both happened. The centurion asked the elders to talk to Jesus and went with them. After they made a formal request, he reiterated what was asked.
Thank you for commenting. It is a refreshing change from the avoidance.
You do understand, I hope, that your version specifically contradicts Luke. See the part in bold.
After Jesus[a] had finished all his sayings in the hearing of the people, he entered Capernaum. 2 A centurion there had a slave whom he valued highly, and who was ill and close to death. 3 When he heard about Jesus, he sent some Jewish elders to him, asking him to come and heal his slave. 4 When they came to Jesus, they appealed to him earnestly, saying, “He is worthy of having you do this for him, 5 for he loves our people, and it is he who built our synagogue for us.” 6 And Jesus went with them, but when he was not far from the house, the centurion sent friends to say to him, “Lord, do not trouble yourself, for I am not worthy to have you come under my roof; 7 therefore I did not presume to come to you. But only speak the word, and let my servant be healed. 8 For I also am a man set under authority, with soldiers under me; and I say to one, ‘Go,’ and he goes, and to another, ‘Come,’ and he comes, and to my slave, ‘Do this,’ and the slave does it.” 9 When Jesus heard this he was amazed at him, and turning to the crowd that followed him, he said, “I tell you, not even in Israel have I found such faith.” 10 When those who had been sent returned to the house, they found the slave in good health.