When I was convincing a friend to watch Yuri on Ice she asked if it was safe for work and I said "yeah it's fine! .....wait SHIT NO, HOT SPRINGS". My friend did watch the show but prudently decided to not try and watch it during work because here in America we still have an awful lot of nudity taboos. And that's one of the hardest things about recommending anime to very casual watchers, there are so many little moments like that (and in YoI those hot spring moments are pretty small) that by now I don't bat an eye at (well, our nudity taboos are a bit, puritanical I'll say) but it's totally something you have to think about when recommending!
So dear readers let me help you out: a few different shows from this past year that you could possibly reach a broader audience with, although slightly different audiences each time.
I would say that this title has the possible broadest reach of all of the shows on this list and that you could show it to basically anyone who's old enough to watch Studio Ghibli films. They might be a bit bored by parts of it but there's nothing even close to offensive in here. Actually I've been describing the show as "the most similar thing to a Ghibli tv series" for a while now, the plotting is totally different from Miyazaki's works (and I feel like it's different from Takahata's works as well, much less melancholy) but there's a general sense of wonder at the low key magic hiding everywhere in the very idyllic Japanese landscape. It's a very charming show for sure and one that only the most stubborn family members would find fault with!
Showa Genroku Rakugo Shinjuu
Alright there's a method to my madness: this is a show I wouldn't have any qualms showing my mom. Now my mom isn't really an anime watcher but she enjoys Ghibli movies, I showed her Time of Eve which she really liked, and she's a general fan of classic movies, period dramas etc. So for someone in that wheelhouse, a period drama revolving around several characters growing up through a tumultuous period and practicing a dying art would totally be a good fit!
A lot of times I don't think that a long-running, episodic series is the best series to try and pitch to a non-regular anime watcher but there are arguments against that! I certainly wouldn't recommend that you try to show Nastume and the Book of Friends out of order, there's enough character growth and reoccurring characters that it would come off as jarring, but here I think the idea of "well, just one or two episodes" would work pretty well! Largely self-contained and feel good, heck if you have someone try this and Flying Witch you can make an iyashikei viewer of them yet!