I'm really not. The fan/viewer reaction to the last few seasons of Doctor Who had nothing to do with content and everything to do with execution. The series hit the ground running by throwing a series of truly terrifying monsters at an audience that was accustomed to hokiness and camp. Alot of running, alot of one-liners, a lot of gadgetry and quick-escapes. Capaldi-Who replaced most of the running with "hold your breath and hide!" and most of the zippy one liners with angst, tears and apologies (literally in Bill's case). That took alot of the fun out of the show, which is where your disconnect with kids comes around.
It isn't "You can't watch this show, it's too violent and gruesome!" it's "You probably won't like this show, it's too depressing."
Between Discovery and Doctor Who, which of these two shows has ever depicted a person being shot to death with a realistic firearm? For that matter, which of them ever depicted a person being literally burned alive?
The only thing that might throw off Discovery's rating is the Klingon sex flashback at the end of "Into the Forest" which could easily be edited down for the squeamish among us. As for depictions of violence, however, Doctor Who makes Discovery look like Sesame Street: for every SINGLE act of violence in DISCO's first ten episodes, I can find at least 5 identical if not WORSE examples from Doctor Who's first 5 seasons.
The only thing Who hasn't shown us is partial nudity (of a woman, at least) which is pretty much all it takes to get conservatives up in arms. Of course, shirtless Matt Smith wouldn't as many British) raise eyebrows as Karen Gillam in a bikini...