But the parts that should make you extra cautious have wires with a lot of insulation so they look thick. those are the 20Kv bits. you also will want to take the time to drain any big capacitors. you can just short the leads or if you are feeling conscientious(and don't like sparks) drain them with a resister(100kohms or so) between the leads.
US TVs are extra spicy because they sometimes have no earth, so if the power is wired backwards (either in the plug or more insidiously from the wall), everything metal inside the set can be live.
If you overvolt the tube (a worry with this replacement board actually) then it can emit x-rays. Most "modern" CRT main boards have a circuit which detects this, but maybe older ones or cheap knock-offs don't.
In short, if you don't know what you're doing, stay away.
I never heard of any kids getting hurt, so it was fine.
Also long sleeves are good, I was recently zapped at the wrist when fixing an oscilloscope.
I think maybe the 20 kV abates rather quickly on its own, because I dont remember using much any caution when unplugging the anode plug.
Some people would still think "it's rubber so it doesn't conduct electricity so I'm fine messing with this TV".
But yeah CRTs are now so rare that I agree that it's very very unlikely for someone without much knowledge to tinker with them. You have to be really an enthusiast to even still own one let alone try to repair one instead of picking up a nice cheap flatpanel.
Personally I'd forgo the gloves (real protective ones are so clumsy it's hard to really do something productive with them) and make sure the caps are fully discharged. Though generally I'd steer clear from this stuff as I know I don't understand it well enough. I came up in electronics during the final years of CRTs so as my knowledge grew they became deprecated and I never really got the full grasp of them.