Does timeless reasoning have any application in situations where your opponents can't read your mind?

It does give different answers for problems like the Prisoner's Dilemma when your opponent is similar enough to you that they will make the same decisions. As you mentioned, it makes an appearance in HP:MoR for similar reasons. There's no obvious application to Death Note, but I think it could certainly be incorporated somehow. If you've seen the film Memento, you might have some idea of what I mean. (I don't want to spoil Death Note because it really is an excellent anime series, so I'm not going to say exactly what I was thinking.) TDT is certainly not essential to rationality but it is very interesting, so it might be worth including in a Death Note re-write for that reason alone.