When speeches are given by the President and Senate and Congress Members, who exactly are they trying to 'convince' that their opinions given in their speeches should be carried out? Who are they addressing that can implement changes? They 'plead' their case but to whom?
They are pleading their case to the American public. If the public supports their cause, it puts more pressure on the opposing party to support that initiative.
Unfortunately(or fortunately, depending on what side you're on), the same Public treats this disgraceful Washington with the disdain it deserves for its lackthereof competence. I don't think Washington gets to plea for anything until it takes corrective actions on its own and improves the situation.
Its not really a mater of "they get to". It is what they are doing. Whether the public decides to pay attention is a whole other matter.
It was FDR who really started this with his fireside chats. He would go on the radio and plead his case to the American people when congress or the Supreme Court were blocking something he wanted done.
Oh sure, they can. But "getting to"=approval. Without our approval, they're merely howling into the wind and its their just desserts for decades of ineptitude. When they ask: Why Trump? Answer: Because the LOT of them are a bunch of incompetent desk-jobbers. Not much would change for them if they returned to being attorneys, waiting for phone calls for cases.
As far as the question in the OP, I gave him the correct answer. If you want to have a debate as to how effective "pleading their case to the public" is or is not, I see that as a different subject. If I were to give my opinion on that subject I would say that it is probably more effective than you believe. It sounds great to say things like "we have had enough of their nonsense and their time is done etc.", but the reality is there is no "we" that are moving in unison. For the most part, people either vote Republican or Democrat. How they arrive at that vote is a personal decision. When a politician from either side is pleading their case in DC, I don't think there is any way that you can speak for the whole and say "we are not listening", and therefore conclude that their pleading is a waste of time. The reality is that their Congressional or Presidential pleadings will sway some people, and others will ignore them entirely. I don't think that reality has changed much over the decades.