It is a grammatically correct sentence and I would report that as a correct answer.
Yes, it is completely natural, and that's what I would have put down myself if the word selection had given me the option.
"A grammatically..." "an" is only used for vowels or words that start with a vowel sound. Then you will have a grammatically correct sentence...
Huy, I can't believe you got away with that for a year.
By the way, huge thanks to the people who put together these lessons and keep them going!
"He wants to go visit his wife should be the correct answer." In fact, I might remove "he wants to go visiting his wife" as correct. I'm not sure if the exact grammatical reason, but the "ing" seems to only work if it's a continuous activity or something like that ("he wants to go walking/hiking/fishing/skiing/sightseeing" vs "he wants to go eat/sleep/make the bed/cook/fold clothes/wash dishes.") I wish I could explain why these two are different. Thanks!
As a native speaker, it's completely natural to me. The first responder was correct.