I have never, in 41 years, heard or read the word "aglet". Thank you!!!
My tape "aglet", hehe now I'm just looking for an excuse to use the word.., is flat and follows the shape of the strap. It extends out past the strap, just like you describe, and works pretty well.
I had also discovered that I was not pushing down nearly hard enough. But it's such a pain, even with that knowledge, that I end up leaving them suspended for so long that by the time I need to thread them again I've forgotten what I learned the previous time...