It seems that "make a visit" is not a common phrase, and I am not sure what is the difference between "make a visit to a friend" and "visit a friend". The following sentence is copied from The Guardian:
A huge 17.5m leisure trips are expected to take place on the roads
between Christmas Day and New Year’s Day, as drivers make visits to
friends and family, although the absence of commuter and commercial
traffic during this period means the roads will be relatively quiet.
Also Ngram suggests that the phrase "make a visit to" is used.
Best Answer
I'm not sure why but, as a native English speaker, 'make a visit' in this sentence just sounds better to me. My guess is that it is because we tend to generalise these actions in newspaper articles, so the action 'make a visit' is referring to all people as a collective noun.
By saying 'drivers make visits', it makes me think that all drivers are making multiple visits throughout the day, whilst the reality is that most people will just go to one place on Christmas day, so it sounds strange to me. But perhaps that is just my experience of Christmas day!