Normally, the question would not be strong, because it's common to find drashos from words that seem to have no discernible connection to the drasha. What made the question more interesting was the Gaon (Aderes Eliahu Devarim 1:12), who says that the reason the Gemara (Pesachim 30b and Sotah 12) associates Bikkur Cholim with the word ילכו is because the tachlis- the purpose?- of Bikkur Cholim is the הליכה, the walking, the preparation. The words of the Gaon are
But the Gaon is telling us even more than that. Bikkur Cholim is under the rubric of Gemillas Chesed, loving kindness to a fellow Jew. Every Chesed has two beneficiaries- the recipient and the provider. The benefit to the recipient is obvious, and the dual benefit to the provider is the mitzva bein adam la'makom and the spiritual betterment that comes of doing kindness to a fellow human being. But specifically in the case of Bikkur Cholim, we ought to realize that beside the obvious benefit to the sick person, and beside the obvious benefit to the visitor's middos tovos, the visitor also benefits personally from the unique spiritual elevation of being in the presence of the Shechina. The Gaon is telling us that, counter-intuitive though it may be, the main beneficiary is the visitor.
In his last years, as my father זצ"ל suffered through his terminal decline, we wondered why such a good and great man would experience such a terrible deterioration. He had saved many lives, he founded yeshivos and schools, he helped the poor, he was a masmid and a world class talmid chacham and pikei'ach. For what it's worth, I decided that the Ribono shel Olam was giving the community the chance to return some chesed, to show hakaras hatov. "Here," Hashem was saying, "this man who has done so much for you, this man is weak and sick and hungry for friendly face. Now you have a chance to show your hakaras hatov." Of course, some people came, even to the last days, and some people moved on to chanfeh others from whom they could derive more benefit and whose association would bring more glory to them. Even among those that came once a year, usually during Ellul, some clearly were patting themselves on the shoulder, thinking "What a fine man I am, visiting the Rabbi in his sad condition." Wrong. You were doing yourself the biggest chesed by visiting. I don't know what the best metaphor is, but those last years separated the men from the boys, they winnowed the grain from the chaff, and they were an x-ray that showed what lay beneath the surface, and it wasn't always pretty.
- Thank you very much, Michael, for sending me the text of the Kli Yakar in Parshas Korach, which, as discussed in the comments, states clearly that the person who is visiting benefits no less than the person being visited. While Rav Schwab talked about the Shechina being there because the sick person re-examines his life and regrets his past, the Kli Yakar doesn't mention the Shechina being there. He says that when people visit the sick or see someone else suffering, they do some soul searching which leads to some degree of teshuva, and this redounds to the benefit of the person who is suffering, who inspired that teshuva. The Kli Yakar says that Moshe Rabbeinu asked Hashem that Korach and his followers should come to an end that is completely outside human experience, so that nobody would look at them and think that such a thing could happen to them for their own sins, and so no benefit will come to the camp of Korach for inspiring teshuva. Implicit in the Kli Yakar, and central to our discussion, is this: that just as the Aseres Ye'mei Teshuva is called בהמצאו, so, too, the bed of a sick person, which is mesugal to Teshuva, is בהמצאו. A time and place that inspires Teshuva is a time and place where the Shechina is more accessible.