^^^^^^^^^^^^^^^ THIS

While it may seem like I am being oversensive about the subject, it is understandable because I left out a large part of the context. When you have heard of this phrase hundreds of times over multiple years by people who were not in the position to help you in the least bit, it becomes a meaningless platitude.

Are you assuming/implying that the people who say "let me know if you need anything" don't actually mean it? Because I don't believe that's generally the case, especially if they're actually friends and not just random acquaintances.

I don't personally say that sentence outside of a work context; usually I would say "call me if you need to talk", but I won't take proactive action unless I believe my friend either wants me to or need me to.

At the end of the day, someone makes a choice to interpret this as a platitude. How about taking up the person on their offer?

It's not for you they say it, but for them and or others. Or to put it more directly, you don't say it for them, you say it for you.

It's like forgiveness, if you forgive someone, the main benefit is to you. By forgiving someone you take a load off your own shoulders. It's often meaningless to them if those forgiven know they are.