GH – are you not aware that saying Kapittel 13 thirteen times does in fact fit into the category of Tefillah? You said it yourself – “there is nothing more effective than direct teffilah from the heart to the Ebeshter. He doesn’t need/want intermediaries or mystical accessories”. In other words, saying Tehillim is a good ‘Segullah’.
There are numerous sources (that all agree to) that there are specific Perakim of Tehillim that are more appropriate for certain times of need. Doesn’t everyone say Perek 16 and/or 91 at a Levaye?
I don’t know the source of this one and never heard of it, but unlike other crazy segulos, this one probably won’t do harm and is muttar.