I don't know how to theorize this but somehow I feel that the only grammatically plausible modification relation here is that "友だちへの" modifies "追加". What am I missing here?
You are missing "common sense" which takes years of experience dealing with exactly these types of sentences, which is difficult to explain logically. Many Japanese people are puzzled with these wordings too, especially those who are not used to such interface. Common sense is not common after all.
Let's break it down:
Firstly, 友だちへの追加を許可(する) can be interpreted in two ways:
- 「友だちリストへの追加」を許可する
- 「友だち」へ対して、「追加」という行為を行う事を許可する
So it's already ambiguous. Here's where the "common sense" comes in to play. Don't ask me how I know this, but 友だち here does not mean (your) friend. 友だち here is pointing to a "list of friends" or "contact list". Actually from an engineer's perspective it's all about attributes - users who have the attribute "friend" appears on "list of friends".
So let's replace 友だち with contact list.
「contact list」への追加を許可 = allow/permit the addition to "contact list".
Now we're getting close.
We still don't know "who is giving permission", "who is receiving permission", "who's contact list". Again let's use "common sense" - you are at the liberty to press/slide the button, thus you have control over permissions. Also, look at the description: 「あなたの電話番号を保有しているLINEユーザー」
So the answer is, that button/slider enables/disables the following:
You allow "あなたの電話番号を保有しているLINEユーザー" to do "addition to (their) contact list".
To make it simple, you are allowing "LINE users who possess YOUR phone number" to add you to their friend list.