Here's a quote from the writer's 7 year old daughter: "These can't be for kids, and comic books are for kids, and kids aren't supposed to see that." I disagree. Comics are a medium, and there's scope for a lot of different content; some stories are suitable for children and some aren't. This is where age ratings can help, although they can only provide a rough guide.
For instance, one of my regular monthly comics is Scooby-Doo Team-Up. I like it: it reads very much like watching an episode of the relevant animated series. Each story is 2 issues long, with different guest stars. They started out with Batman, then moved onto other DC characters, and most recently they've met the Flintstones and the Jetsons (via time travel). So, I was thinking about getting a subscription for my 7 year old nephew, until I noticed that the age rating is "9+". I haven't seen anything in there that I'd consider unsuitable, but I'm not a parent and I may not be the best judge of what a young child would find scary. Anyway, it only costs 69p per issue, and you can read issue #1 free via Comixology (link above) if you want to try it out.
Coming back to the other blog post, the writer said that there were loads of Batman/Superman/Wolverine comics, so he told his 5 year old son to go and pick 3 of them. I think this may have been a bit of a hasty decision.
I wrote about DC's "New 52" back in 2012, where they rebooted their continuity. I picked up a few new titles, but most of them have now finished (Animal Man, Demon Knights, Resurrection Man) and I gave up on Batgirl a few months ago after it got a new creative team and a new direction; the only "mainstream" DC comic I read now is Swamp Thing. Anyway, a few characters were redesigned during this reboot: the writer mentioned Harley Quinn (I'll come back to that), but the Joker has also changed. In particular, someone cut his face off and he then stapled it to his head to wear it as a mask. (See THE STRANGE SAGA OF THE JOKER'S FACE for more details and some of the artwork.) Would you want a 5 year old to read a story like that? Obviously the writer knows his son better than I do, but I'd be hesitant.
So, it's not as simple as looking at the costumes on the cover: I think you need to consider the story content as well.
I tried out a new series earlier: The Unbeatable Squirrel Girl. (Issue #1 was published last week.) I have to say, I really enjoyed it. It's a lot of fun, and I sat there with a big grin on my face as I read each page. There's some sample art etc. on the creative team's Tumblr, and you can read the first few pages on Comixology. Again, I think that children would enjoy this; although it's set in the main Marvel universe, you don't need to know any backstory to follow what's going on here. However, checking Marvel's website, I was surprised to see that they've rated it "T", i.e. aimed at ages 12 and upwards. (Marvel don't actually describe their rating system on their own website, but there's a list on Wikipedia.)
Sticking with a teenage audience, I recommend Kelly Sue Deconnick's work on Captain Marvel (2012 and 2014) and Avengers Assemble. Ms. Marvel is also worth a look. As I mentioned, I gave up on Batgirl recently, but I think the new direction is specifically aimed at teenage girls, so they might get more out of it.
What about pre-teens? Marvel have some all ages titles which use "Marvel Universe" as a sub-brand, e.g. Marvel Universe Ultimate Spider-Man (not to be confused with Miles Morales: Ultimate Spider-Man). I haven't read any of them, and I think they might be direct adaptations of TV episodes rather than new stories, but it's somewhere to start.
Over at DC, you could look at Tiny Titans. That series has now finished, but the paperback collections should still be available, and there's a new series (Return to the Treehouse) coming in June. I haven't read that, but I do like the fanfic comic JL8 (available on Facebook and Tumblr) which has a similar theme.
The other blog writer specifically mentioned Harley Quinn. Leaving aside the merits of her new costume, I haven't read any of her stories since the big reboot. However, I did like the character in Batman: The Animated Series, and it's a pity to lose such a distinctive costume. Still, even if the current comics aren't to your taste, older stories should still be available. On that note, take a look at Top 5 Harley Quinn Comics To Read.
It would certainly be good to have more of a range available. When I was young (in the 1970s/1980s), I started out with the Beano and then moved onto the Eagle. There were also lots of other British comics available, which are all long gone now. Cancellation is still a problem for American comics; I think that the average readership is about 20,000, which is roughly the same number of people you could fit into the football stadium at Selhurst Park (to watch Crystal Palace play at home). So, the best way to get more diversity is to vote with your wallet, i.e. buy the comics that you like.
As for comic shops, there may be an argument for changing the layout so that comics for young children are separated from the rest. (That seems to be a common idea in libraries, with a smaller room for children's books.) In London, Gosh! do a good job with that: there's a set of shelves near the main entrance aimed at a younger audience, then you can go downstairs for the full range of monthly issues.