I used to think one could get by without assertions in a society, using only promises. Here's the trick I had in mind. Instead of asserting "The sky is blue", one first promises to utter a truth, and then one utters (without asserting!) "The sky is blue." This has sufficiently similar normative effects to actually asserting "The sky is blue" that a practice like this could work. This observation could then lead to further speculation that promises are more fundamental than assertions.
But that speculation would, I now think, be quite mistaken. The reason is that we do two things with a promise. First, we create a moral reason for ourselves. Second, we communicate the creation of that moral reason to our interlocutors. Both parts are central to the practice of promises: the first is important for rationally constraining the speaker's activity and the second is important for making it rational for the listener to depend on the speaker. But communicating that we created a moral reason is very much like asserting a proposition--viz., the proposition that we created a moral reason of such and such a type. Consequently, promises depend on something that, while not quite assertion, is sufficiently akin to assertion that we should not take promises as more fundamental than assertions.
A similar phenomenon is present in commands, requests and permissions. With commands, requests and permissions we attempt to create or remove reasons in the listener, but we additionally--and crucially--communicate the creation or removal to the listener. Assertions, promises, permissions, commands and requests seem to be the pragmatically central speech acts. And they all involve communicating a proposition. Assertion involves little if anything beyond this communication. In the case of the other four, the proposition is normative, and the speech act when successful also makes that proposition be true. For instance, to promise to do A involves communicating that one has just created a moral reason for oneself to do A, while at the same time making this communicated proposition be true.
So something assertion-like is involved in all these pragmatically important speech acts, but they are not reducible to assertion. However if we were able to create and destroy the relevant reasons directly at will, we wouldn't need promises, commands, requests or permissions. We could just create or destroy the reasons, and then simply assert that we had done so. But our ability to create and destroy reasons is limited. I can create a reason for you by requesting that you do something, but I can't create a reason for you by simply willing the reason into existence. (Compare: I can make a cake by baking, but not by simply willing the cake into existence.) However I can create and destroy reasons through speech acts that simultaneously communicate that creation and destruction, and that's how promises, commands, requests and permissions work.