I think the structural value of proteins comes from the diversity of amino acids (although possibly causality went the other way there). Lets consider all our possible structural elements. They need to be something that's big....basically, they need to be a chain of repeating units that's arbitrarily long. Fats don't really do this.

That leaves sugars, nucleic acid, and protein. Long sugar chains are sometimes structural (see cellulose and chitin) but there are only a few kinds of sugars used in these structures and they don't have a lot of complexity in chemical nature. So the number of things you can make from them is limited. Nucleic acid has more options, with four bases it can and does bind to itself to form special molecules that use special structures that have catalytic function. A lot of people think this used to be what life originally used, before proteins really got going (the RNA-world hypothesis).

And finally we have proteins. Proteins have 20 types of amino acids to work with, which allows for a ton of variety in chemistry of each individual part. Which in turn means proteins can do all sorts of things, because the different arrangement of parts let proteins fold in all sorts of ways and interact with their environment and each other in all sorts of ways. With so many different amino acids, you can make a ton of different proteins that do a ton of different things...including structural proteins.

To make an analogy, building with sugars is like if you have one kind of lego block, building with nucleic acid is like if you had 4 kinds, and building with protein is like having a 20 different widely varying lego pieces to work with. It's just more versatile.