Musar על בבא מציעא 172:11

It is obvious to me that the מצוה of being hospitable to guests, (which is certainly included in the overall commandment to follow in G–d's footsteps), represents an additional virtue. The Torah has singled out this story of Abraham personally performing chores to entertain these strangers. It also alludes to the reward accruing to him, i.e. that everything he did personally for these guests, G–d later on personally did for Abraham's descendants, whereas whatever Abraham had designated to be done by someone else, G–d also performed for the Jewish people only by means of one of His agents. You should not think that all we talk about here is merely a part of a virtue. Whoever performs such an act such as Abraham's, performs the positive commandment of emulating G–d, and he will receive the reward due to someone who has performed a מצוה which has been specifically commanded. G–d performs acts of hospitality at all times, at every moment. Were He to desist from doing so for even a single moment the world would be lost. All creatures are His guests, we are all like strangers vis-a-vis G–d, and just as strangers we need a place to sleep. G–d enables us to survive. If He were to withdraw life-support for even a moment, we would all perish.

This also explains why this act was performed by a messenger and not by Abraham personally. Abraham represented the essence of that virtue, i.e. מים רבים. It would not have been appropriate for Abraham also to personally perform this secondary function of the waters. Even G–d provided purification for the people of Israel through an agent, i.e. the waters which came forth after Moses struck the rock, as distinct from the manna which G–d Himself supplied from the Heavenly Regions. We are taught here that man should do good and not need to be purified by waters except after he has committed sins. When man lives up to these criteria he can achieve דבקות בה' on what is known as a פנים אל פנים "face to face" basis rather than through an intermediary. This very characteristic is described in Job 28,25 as ומים תכן במדה, "He set the measure of the waters."