Whoever initially suggested discharging CRT's for typical VDU maintenance, unfortunately got it wrong. Ignore the youtube videos suggesting this should be routine.
The total amount of charge stored in the CRT bulb is not enough to harm you, it gives less of a zap than a typical farmer's electric fence. The only risks are if you take off the anode cap (EHT cap), and get a zap, you could withdraw your hand rapidly and cut it on a sharp object. Or you could, if carrying a non discharged CRT across the room, get a zap and drop it on the floor.
In general VDU servicing there is no need at all to go under the insulated anode cap, unless you want to test the EHT with a probe, which you slip under there with the set running and generating EHT, or , with the set off, you need to remove the cap to replace the LOPT or the CRT.
Unless the insulation of the cap and EHT wire is grossly degraded, you won't get a shock from it, so simply leave it alone. You are more likely to get a zap going under the cap to attempt a discharge when your earth clip falls off.
Also, when you do have to remove a CRT and/or Lopt, and want to discharge the CRT first, do via a resistor to a solid ground point, a 100k to 1M 2W type is fine. This limits the peak discharge current (that you get with a direct short) which can actually be in the hundreds of Amps vicinity, otherwise in some cases can cause damage to the internal Aquadag connection to the metal anode connection on the inside of the CRT bulb. An even better method that I use is simply to slip the EHT probe tip under the cap with the set turned off and the resistor in the probe, typically 100M Ohm, gently discharges the CRT.
The other option, prior to removing the anode cap, if you are shy about the discharge, just wait a few days and the charge mostly dissipates, this is because the semiconductor EHT rectifiers do not have zero reverse leakage. This was not the case with tube EHT rectifiers though.