Hello,
the 555 timer (low power versions are now readily available too) is the classic way to do it precisely with few extra components, the uC
+MosFET is another interesting way, especially if you would like extra functionality and already have a uC programming setup.The direct extension of your initial method is a transistor to buffer the RC timing circuit and let you use a smaller capacitor, however the gradual turn off problem will remain, and the component count is not less than the other two methods. Relatively expensive and large timeout relays also exist with a dial to set the timeout period.
In either case *do not omit* the R/C/D snubber circuit (typically about 100ohms in series with 100nF, plus a 1A diode in parallel or some resistor series combination, as close to the relay coil as is practical) unless you are really certain that they can be omitted in this case.
The snubber circuit will protect both your own circuit, and others. Inductive relay circuits often work on their own without them, however they are now substantial radio frequency generators and can stop other circuits/devices from working when connected or placed in proximity, even a device that is not introduced to the room until years later.
I once made a waterpump circuit switched by a relay with an R/C/D snubber mounted beside the relay coil. This worked well until I introduced an accessory to the same control computer connected by the (supplied) unshielded ribbon cable. It took a while to figure out that the program was crashing because of RF feedback, even with the snubber circuit, to the computer accessory memory via the unshielded cable. Grounded tinfoil on the cable provided a complete temporary fix.
Three bathroom 2kW down-flow heaters have now experienced exactly the same early brownout failure in my house. An examination of the pull cord switch shows that a plastic pin in series with the the pivoting metal plate (showing substantial arcing) gradually melted, shortening each time, until the switch no longer worked. The AC inductive heater coil has no R/C, or R/C/D snubber circuit. I am replacing the pull cord switches with fused wall panel cooker switches (wall panel fuse plates are already present) and will add R/C (available as in single packages for power loads), or maybe even R/C + back-2-back zener diode snubber circuits considering the AC operation in this case.
DH