Second, my feeling is if someone sits down to think about an issue, writes a letter and stamps their name on it, then they probably become more informed and involved in discussion than another guy who just ticks a box on a petition. This informed person can go on to discuss the issue with friends or form his/her own local group. So, writing letters to senators is useful, even if it is just a first step or means to an end.
Nobody needs to change their opinion as a result of my letter. I wouldn't respect them if they did so without consideration. So (2) is irrelevant.
I didn't intend to clash with you, I'm just stating my opinion which is a bit different than yours. It's perfectly fine if you want to say writing letters is useless. I feel differently.
If you sit back and assume your district is already modeled, etc etc, then you are not participating in democracy and you have not seen for yourself who else out there might agree with you. The most important things about democracy are your vote and your voice. One without the other holds much less strength.
(1) If your district isn't contentious, that means the sum of factors aren't of sufficient effect size to care about that specific district. With limited funds, you concentrate elsewhere.
(2) Just because there's noise doesn't mean that people's opinions have changed. You're assuming that people's opinions have not already been modeled. Some elected official may suddenly get a flood of emails about gun rights in proximity to some highly publicized event on gun laws. That does not mean that the distribution or magnitude of opinions have substantively changed.
(3) Some groups do have negotiating power because they're aware of the political calculus more intimately than the average noise-making group like Black Lives Matter. They're able to distribute distilled information to their members and organize effective get-out-to-vote drives in the districts that matter (meaning they also do not wantonly spend their finite resources).
If you don't organize, then you are noise. Organizing is more important than voting (or writing a letter) itself, because part of organizing is dealing with finite resources through the skillful attending to the places that matter, and skillful neglect of the places that don't matter, implying that you can actually deliver on a get-out-to-vote drive when the time (and places) matters.