2024 Lich and Anti-Magic Field, how boned is the bone man?

I've got some players wanting to find a way to have anti-magic field on my campaigns BBEG and I was curious what the ruling is on what part of the Lich's stat block is affected. 

Obviously none of the spells would be cast while in the AMF, and his legendary action that casts fear would be a no go as well. But both of the Lich's attacks are listed as melee (or ranged) attacks does that mean they can be used while in the AMF field since they aren't listed as being magic attacks or spells? 

Or since basically everything a Lich can do is magical in nature, does that mean attaks are only going to be unarmed attacks?