How does Islam view it when many people come to visit you?
With joy. Every visit is a gift, and friendship is gladly returned more generously than it was offered. Only limited time sometimes makes it necessary to share visits out fairly.
In Islam, visits are something beautiful. Whoever goes to see another person gives them time and affection. Whoever receives such a gift is glad and wants to return it. The verse above teaches us not only to return a greeting but to answer it with a better one. Friendship is the same. We gladly answer affection with even more affection.
Such meetings give strength. Receiving many people does not drain you. It lifts you inwardly. When whole families come and the children simply sit down with you, it feels like one single, large family. Whether the guests come from Africa, Asia or America makes no difference. They all belong to the same worldwide brotherhood.
There is still a limit, and that limit is time. A day has only so many hours. If some people come very often, others may get no chance at all. Then it is fair to ask kindly for understanding and to make room for others. This does not mean you are unhappy to see someone. It only means that nobody should miss out.
In Islam the guest holds a place of honour. As the hadith above shows, honouring a guest is part of faith. So whoever receives visitors may rejoice and should honour them. And whoever comes to visit keeps in mind the others who are waiting too.