I think Naruto's surname is not Minato because he wants to protect him. There are two other reasons. Let me explain in detail the reasons why Naruto's surname is Uzumaki and not Minato in "Naruto":
First, when Naruto was first born, his parents died in the Nine-Tails War and did not have time to give Naruto a surname. The name Naruto was decided by Jiraiya and Minato long ago. Well, after his parents died in the war, Minato made many enemies. In order to protect Naruto, the third generation inherited the whirlpool to Naruto. In the early days, even Naruto didn't know that the fourth generation was his father.
Secondly, Minato was an inverted gate. At that time, Minato did not become Hokage, but the Uzumaki clan had a good relationship with the Senju clan and had a higher status. In order to continue the inheritance of the Uzumaki, Minato was allowed to inverted.
Thirdly, Minato is afraid of his wife. This reason is not difficult to understand, because Minato and Kushina have a good relationship. Kushina usually has the upper hand in family status, and Minato just silently protects Kushina. Shinai was completely obedient to Kushina, so as a wife-loving Minato, he didn't care much about Naruto's last name.
The above three points are why I think Naruto in "Naruto" is named Uzumaki and not Minato. Whatever the reason, I don't think this arrangement is necessarily bad.