Is there really any safety concern (beyond the obvious), regarding putting 240vac mains into a basic potential divider that is 210k Ohms?

• This circuit does work.

• If the 10k resistor fails open (and resistors do normally fail open), nearly-full 240V goes to the output (or a higher voltage depending on load), probably blowing it up.

• Both output wires should be treated as live wires, for the purposes of safety. You may think that by connecting the output near neutral, they don't have to be treated as live, but consider that if someone brings your circuit from the UK to continental Europe and plugs it in with an adapter, live and neutral may be reversed.

• Because of the above, this can only be used if everything downstream of the power supply is insulated. It's not clear what "a doorbell" means to you. If this is going to connect to a doorbell button with wires (you mention replacing a doorbell transformer) that's not safe because the whole doorbell system would have to be treated as live wires. If it's all contained in a plastic box, and you're just receiving a wireless signal and playing a dingdong on a speaker, it could be okay.

• Also you must not let them touch anything else earthed. Measuring the output on one channel of your oscilloscope carries a risk of blowing a fuse (or worse) on your oscilloscope. A differential measurement is required, since the output isn't reliably earth-referenced or floating. And of course, avoid touching the things you're measuring.

• Each 100k resistor dissipates about 130mW continuously. Make sure to size them adequately. A typical 250mW resistor may be technically able to handle 130mW, but it will get rather hot, possibly hot enough to melt some plastic. This is why, when a voltage divider circuit is used like this, a capacitive divider is preferable to a resistive divider.

  Burning under a watt, continuously, isn't that much power, but it can add up if you sell a million of these things and they're plugged in all the time, especially because that power is serving absolutely no purpose. There've been initiatives in different countries to try and reduce standby power.

• Capacitive voltage dividers are used in real products (cheap ones) so evidently, it's acceptable to use this type of design in the real world. Such products are double-insulated so nobody can poke any of the circuitry inside and get electrocuted, even if live and neutral are reversed. The principle of a capacitive divider is pretty similar to what you have here, except with capacitors, and it only works on AC.

  Just like a resistive divider, the capacitor size should be chosen to provide the needed output power without too much idle current. Unlike a resistive divider, the current is out of phase with the voltage so the unused energy is returned to the mains each cycle instead of being turned into heat.

• Compliance requirements are out of scope of this answer because I don't know anything about compliance.

You have used the term ‘ELV’ in the diagram. Not a good idea, as that has a specific meaning in some jurisdictions (and it is likely wrong).

In fact both sides of that output require the same very reliable high-voltage insulation from any human contact as the 240VAC mains itself, due to the risk of current flowing through a user to earth ground. Typically this will be “reinforced” insulation (sometimes called double insulation). Human danger and severe damage from a similar cause could also result of this circuit was connected directly to some other circuit such as an oscilloscope or computer.

You could use an isolation transformer to provide galvanic isolation, however they are generally heavy and expensive since they operate at mains frequency. Here is a 120VAC type that I use myself to allow safe testing of “live” circuits such as the one you’ve shown. A mains step-down transformer would be a better solution if you want low voltage AC. You could apply a voltage divider to the low-voltage output. Here is a 120VAC wall plug adapter that performs a similar function at a lower mains voltage and with a North American plug.

If we assume that the components shown are locked into a properly insulated enclosure that cannot be accessed without intentional use of tools, then there still remains risk of fire or drawing excessive current that could damage the mains cord. A fuse rated to protect the external wires will mitigate the latter. Resistors such as carbon film can fail in a low resistance mode that does not blow the fuse but produces a lot of heat and smoke.

As mentioned in another answer this has similar safety concerns to that of a “transformerless” capacitive or resistive dropper power supply used some home appliances such as, for example, a kettle where the function keys, heater, display, LEDs etc. cannot come into contact with the user and special “fusible” resistors are used that fail in a predictable and safe way. Such circuits can be made safe enough in such very limited circumstances, but it is far from inherently safe, and generally the rule that we espouse here is that “if you have to ask about transformerless supplies, don’t do it”.