Is the only option either for everyone who needs another socket to get an adapter and charge things in their own rooms or constantly swap which appliance is plugged in? I live in a shared rental where everyone has different brands of electric toothbrush and shavers so even a single extra plug would be helpful but plugs without this shutter effect seem impossibly hard to find.
Those shaver sockets are actually a small transformer with 1:1 windings. That provides mains voltage in, and the same voltage out. The only difference is the output voltage is "fully floating" with absolutely no ground return or reference. You could accidentally touch the voltage and the faucet with no current flow. The only way to get a shock would be to touch both sides of the output.
They are an invention from a time when there was a lot less electrical products and before the time of electric toothbrushes. They were designed for powering electric shavers only, as the name states. Their use has morphed over time to now charge devices. However, you are not supposed to use two at the same time. They are designed to power a single use item at a time. The type of transformer in them is weak and will not power heavy duty appliances. Or using two at the same time. If you have multiple items then you will need a second shaver socket. A rental location is not an ideal scenario for them by your own admission. You can get chargers that will allow these products to be charged in a standard socket outlet.
Please do not be tempted to bring in an extension cable from outside the bathroom. This is very dangerous. The wiring regulations do permit any standard sockets in a bathroom because of this danger.