Because of the very emotive nature of this argument, there are a lot of surrounding factors brought into it, such as, "We died for the right to vote." "It is your responsibility and civic duty to vote." And so on. I've heard them all! Including the "You get the government you deserve." But that is really just an ad hominem attack against you rather than having any merit in logic, since your vote has nearly zero influence on the government that comes into force. How do you deserve what you have no influence over?
The many biases we have on this prevent logical reasoning on the matter, in my view. So let us strip the labels out and instead talk about bets. Some people like to do the lottery and some don't. In either case, the odds of winning the jackpot are tiny and in the long run most people lose. I do not hear emotive arguments complaining about not doing the lottery, yet it amounts to the same as voting in elections, where the jackpot is having your single vote altering the outcome of the election. In fact, because I have the courage of my convictions, I am voting Labour in the UK at the next election, even though I am a Conservative.
Regarding keeping quiet about politics if you don't vote, I don't see how that is relevant. If you enjoy discussing politics, why does not doing a futile thing (voting) mean you shouldn't talk about it? In fact I would argue that your individual vote has a net negative effect on the world. You contribute to congestion, global warming if traveling by car, risk of accident, consuming resources and so on.
This is not to say I do not believe voting is a bad thing. We need it. Please don't confuse what I am saying with what you think I am saying. My argument is simply that your personal vote does nothing. You are kidding yourself if you think it is. It is a paradox of sorts. Take some time out to think about it. If you disagree with me, tell me how many elections you have changed the outcome of in your lifetime. If you don't want to answer that, tell me how many elections in the UK, USA, France, Germany, Spain, Italy, Australia, South Africa, Holland, Sweden, Norway, Switzerland, Finland or Portugal have been changed by one vote in the last 100 years? Maybe therein lies your answer.
American humorist Mark Twain is quoted to have said "But in this country we have one great privilege which they don't have in other countries. When a thing gets to be absolutely unbearable the people can rise up and throw it off. That's the finest asset we've got -- the ballot box."
Yes I agree, when the people rise up to vote against it, that is a good thing.
Keith Ellison said: "Not voting is not a protest. It is a surrender."
If an individual doesn't do something that doesn't have any material impact, it is not a surrender. Keith Ellison is only right if he is referring to groups of people voting, not an individual. Once again, it is a paradox.
President Franklin D Roosevelt is right to say what he is saying, not because an individual's vote does anything, but rather that persuading a collection of individuals is more likely to have an effect, and that is the task of politicians.
There is a lot of motivated reasoning behind this issue where people are not prepared to think of the logical ramifications that one vote actually has. It has become an ideology that your one vote counts, because everybody perpetuates the myth again and again. Consequently, rational analysis of this by most seems very difficult. If no one wants to do the math on it, what does experience say? I think the real world is on my side with this one.
P.S. Don't shoot the messenger!