How can I add more "chew" to veggie burgers with only vegetables?
Vegetables are mostly water. Water isn't chewy, protein is. What has a lot of protein in the vegetable world? Legumes, grains and fungi. Your "usual substitutes".

I feel like the requirements that you have laid out are self defeating.

Instead of not allowing the known chewy foods you should instead figure out how to use them for bulk and still showcase more flavorful vegetables.

I prefer white wheat that has been boiled until it bursts. Lots of texture but the flavor is easily overridden by whatever else is added.

As for a binder- the easiest is an egg if you don't need vegan. Otherwise you can use starch such as flour but you have to be gentle.