It sounds like they need a "Don't Bug Me" setting...
No ads huh?
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Yes, no ads simply means they won't serve you up ads that chess.com gets paid revenue for. But you will still get chess.com's own promotional contacts.
Just use the iPad app no ads ever
So last week I tried the premium membership out, and it was pretty pointless. I really wanted it for the no ads, i understand chess.com needs money, I do, but they're annoying. So yes, some ads were removed, but very few it seems. I kept getting recommendations to follow this person, follow that streamer, I got things telling me to do tactics, lessons, drills, vision, chess tv, that's not getting rid of ads. Its the same kind of annoying stuff, so what's the point in a membership? And is there another way to get rid of this stuff?!
Those things are not considered ads; they are not trying to sell you anything just make you aware of things to do on the site.