If you wish to know what part of speech 本当に is according to "school grammar" in Japan, then it is a [副詞]{ふくし}. Every kid in Japan is being taught that it is a 副詞 as I type this. I want to stress that using English grammar terms such as "adverb" and "adjective" will only slow down your Japanese study in the long run, trust me.
There are a handful of people who claim otherwise regarding 本当に, I admit. So the choice is yours. You could either join the main school of thought or go independent. After all, words never come with tags telling what parts of speech they belong to. You will need to decide which theory makes more sense or seems more persuasive.
I learned English in Japan by using whatever books that happened to be around me (and there were only a few). Years later, do I look like I know nothing about English grammar? Do you have difficulty in reading my English?
You will surely need to know what types of words 本当に can modify in order to use it correctly, but whether you believe the word is a 副詞 or not will not make a cool topic at the party. Cheers!