I was approached by one of the "big publications" you mentioned and specifically asked for one of the articles I had published the day before. I submitted it, it was accepted, and it was nowhere to be seen, right from the outset.
If you request a story — one that has never been submitted to you before — it would be good to let writers know this reality. Or to feature the story, since you requested it.
I’d rather publish with Writer’s Blokke, Benny Lim, than with most of the ones you mentioned here. There are also some really good, smaller-to-midrange publications where Editors really do edit (it’s a labor of love, and they’re volunteers, so you’re really lucky when you get a good one). I’d aim for those over the “big” ones. That way, you know your work is rubbing elbows with other quality work.