To understand why anyone might think speech acts are important, it's important to understand where the idea of speech acts appeared and what it was in response to. Speech acts are as you correctly tag part of the philosophy of language.
As the story goes, the philosophy of language is a discipline that picks up in the late 19th century which sought to (a) clarify the meaning of concepts, (b) abandon metaphysical claims, and (c) achieve greater precision in argument and logic. A big part of this in the English speaking world is a now unthinkable reality: the dominance of Hegel at Oxford (famous names: Bradley, McTaggart). Bulky sentences and talk about Spirit were the norm. In response to this, people like Russell wanted to do the polar opposite.
Th German Frege wrote an important paper in 1892 called "On Sense and Reference" which was crucial for both logic and the philosophy of language. The important point is the following posit: every sentence has sense and a reference. Translated into what it evolved to: Every statement has a meaning and a truth-value. In other words, everything can be reduced to sentences that are either true or false.
Conversely, any thing that does not have a proper sense (i.e. description of something in the world that can be true or false) is "non-sense." Taken carefully, that's not something terrible since "non-sense" would be merely a technical term, like "heretic" in some bygone era. But if sense/non-sense is not just about a limited part of language but reflective of whether there's any meaning at all in things, this suggests morality, theology, etc. are bunk. Combined with a strong dose of reductionism and you've got a sad world where: "I love you" just means "When I think of you, a certain chemical passes through me" = TRUE, and there's no such thing as right or wrong because Good does not relate to whether something obtains in the world.
Austin and later Searle in developing speech acts are positing that this account is missing another very important function of language: language is also performative. It accomplishes things. Austin's examples:
Austin’s opening list of examples of putative performatives includes: “I take … to be my lawfully wedded …”—as uttered in the course of the marriage ceremony; “I name this ship the Queen Elizabeth”—as uttered when smashing a bottle against the stern; “I give and bequeath my watch to my brother”—as occurring in a will; “I bet you sixpence it will rain tomorrow” (1962b: 5). (SEP)
What important philosophical problem does this work on?
First, it suggests a solution for what to do with sentences that didn't seem to reduce down to their sense.
Second, it chips away at the previously existing pattern (I don't know if anyone claims it explicitly) that all language is reducible to either statements with a sense and reference or non-sense that shouldn't be used for doing philosophy.
Third, it has application in understanding Chinese philosophy. Fingarette's Confucius: The Secular as the Sacred is largely an account of how Confucian Li (禮) is speech acts that show respect and reverence. Admittedly, that's going to primarily be looked at people writing in ethics, but the point in part is that ethical claims are not merely disguised sense-reference claims, but rather involve thinks like action and intention in ways that suggest for many that they are metaphysically real.
Gricean implicature similarly suggests that there's more going on in our use of language than the meaning of the words themselves. Again, this is an attack on the notion that all there is in language is sense and reference.
Put another way, to process anything linguistic takes more than the ability to boil utterances down to their refined sense and judge whether or not they are true. It involves a complex set of mental and cultural capacities that enable understanding, and it's only a trivial subset of sentences that can be fully grasped with the meaning their speakers intend by merely grasping the function of each word and part in achieving sense.
In both cases, this points to and partially resolves an incompleteness in philosophy of language's account of language, and if philosophy of language can't give us an account of language and how it works, then it itself isn't of much use to philosophy.
Finally, you make a further point that philosophers in doing their work express themselves using assertions. I think that's basically true or it has been in my experience and I shudder to think that my own work wouldn't be largely composed of claims and arguments for those claims.
But then when I think about it, isn't that the nature of doing philosophy rather than a demonstration that speech acts and implicatures aren't worthy objects of study in the philosophy of language? As in, we write in a certain style in philosophy, because the purpose of publishing in philosophy is precisely to make and defend these sorts of assertions and thereby provide clarity on whatever domain we use them in. Conversely, most of the rest of life involves using language in a more normal fashion where much of what we say and write does not match the mold of this method. (This reminds me of a quote on the door of a professor back when I was an undergrad:
“I am sitting with a philosopher in the garden; he says again and again 'I know that that’s a tree', pointing to a tree that is near us. Someone else arrives and hears this, and I tell him: 'This fellow isn’t insane. We are only doing philosophy.” (Wittgenstein, On Certainty, quoted at https://www.goodreads.com/quotes/113614-i-am-sitting-with-a-philosopher-in-the-garden-he )
with the point being that the narrower confines of language usage and the odd things we bother trying to say and prove in philosophy are a horrible basis for deciding the limits of what philosophy of language needs to consider.
