I've delved into the subject a bit and as someone else has mentioned, "homograph attacks" are a thing.
Looking deeper, it seems each TLD has a set of dictionaries and rules for what is an acceptable combination of characters, and what isn't (to avoid homograph attacks)- and as I understand it, the solution is on a per TLD basis, as in different dictionaries and rules per TLD.
For 3rd parties like LinkedIn, I could understand why they may be averse to embracing IDN domains but for a company of their size and reach, they really should be looking after their wider user base.
For working with IDN domains, I've felt the easiest way to store/process them is to store them as their ASCII value and convert for human consumption.
Tried hunting for a C library that deals with the dictionary issues and was left not entirely sure whether libidn supports them.