It isn't nearly as dangerous as you think it is. People stress the dangers because it is theoretically possible to recieve a lethal charge from the picture tube of a TV that isn't plugged in. Note the word "theoretically"? I don't know of any cases of this actually happening, ever.
I took the full high voltage from a RUNNING vector monitor a few weeks ago. I didn't even get a headache afterwards.
All modern monitors have auto discharge circuits. These circuits USUALLY work. Most monitors I discharge do so silently because they have nothing to discharge in the first place. The only reason you even discharge is because the auto discharge circuits might be bad.
Secondly, the anode cap doesn't just fall off, it would be bad news if it did, but it would be the exact same bad news that would happen if the anode cap fell off a tv set that was inside the case. It would still be a freaking lightining bolt and the monitor would be ruined and probably anything hooked up to it.
Matter of fact not only does the anode cap not just fall off, it is usually quite a ---smurfette--- to get off in the first place, especially if you don't know the right motion to use.
So as far as the anode cap goes you pretty much have to make a DECISION to rip that sucker off and stick your hand in there. It doesn't happen on accident. My incident was with a vector monitor, which is a completely different beast (on a Vectorbeam monitor there is a spring which goes across the back of the tube, this is ground for the tube. I had one with that unhooked, and powered on, managed to touch both the back of the tube AND the frame at the same time, so the tube (which wasn't powering up because it had no ground) was able to briefly power up by going to ground through my body. Until I yelled monkey klaw and jumped back.
Frankly you are more likely to mess up your AC wiring and cause a fire than you are to get hurt by a monitor when you have read about monitor safety. I basically all comes down to two things.
Don't rip the anode cap off and stick your finger inside.
Know HOW to discharge, and WHEN you should do so. (You only discharge if you have to remove the anode cap).