Yah know, if an anime can manage to give me "feels," then it has to be doing something right. Even if it is as contrived as Anohana, it still works through good writing. We all know there are a ton of "try-hard" series out there that just doesn't feel right or work, but Anohana made it work.
Well, here's the thing. Whether or not you are aware, when you're watching a work, it (like every other work) is crafted to bring up all sorts of emotions, difference is in how they do it. There are many ways to approach but ultimately it's a choice. You can start off with a scene where the MC's mother or daughter dies, have them weep and end there - I won't say no one will cry/feel sad but that surely doesn't excuse the writing as good because it made *someone* cry/sad. Basically anyone can write that bit of emotional drivel. I can. You can. Okada can.
Secondly, how susceptible one is to emotional influence differs from person to person. Entertainment often ends up being subject to emotional drain, average guy doesn't want to analyze the work very much (hey did they just add that scene to emotionally mainpulate us? was there any concrete point in the whole show? did <something> have more meaning that it appeared? was there a reason they didn't they do <something>? how can so many things happen?) - as long as it has loosely 'believable' reasons, it's fine. I don't really blame anyone for that, there are many reasons for people to be the way they are but I do encourage going a bit further than seeing a work as passive entertainment, if not always, sometimes. Make the most of the time you spend on something. That being said, there are situations where being overly analytical does deter from the experience.
Good writing comes in when you build on emotion while maintaining logical/analytical consistency - I say consistency because the logic of the world is subject to artistic/thematic choice. No show is perfectly consistent though, it comes to how close they can be and again, artistic choice (which in most cases is explicit). Some might follow a realistic approach with realistic consequences to situations while some might try to exaggerate something. Not going to say good writing will always mean better reception, there are other factors that come into play: themes, direction, presentation etc which again are subject to personal taste. I personally find that strongest emotions emerge from more subtle and understated drama - Shouwa Genroku Rakugo Shinjuu is a shining example of that, it's pronounced knowing that it is surrounding art. I still find myself getting very emotional watching the first and last episode.
(okay, I wanted to make this brief but it got longer than I expected, let me know if I said something wrong).