As per my knowledge, the overload current will trip a circuit breaker and will burn the fuse of any fused switch. How come when my main breaker failed (because of the steamer I reckon because it happens every time we use it) the fuse of the switch that held steamer socket + the ELCB didn't trip (It jumped directly to the MCB)?
"Circuit breaker coordination" is the term used when setting up multiple breakers / fuses. The trip curves overlap and in many cases a small breaker can take longer to trip than a large breaker.
One example I can provide is this: the plant I work in was not built with the same brand breakers. In one area an 800 amp main will trip out prior to the 400 amp mcc feeder or the 100 amp breaker on the motor overload bucket. The cost to coordinate these breakers to the correct ones would be several thousand dollars and it happens so infrequently that the owner is OK with the occasional down time. By comparing the curves of the different types of breakers you can see that fuses and small inverse time breakers will hold at 4-6 times their rated value for a short time.
So depending on the loading of the circuit and type of breakers used, if not well coordinated, the main will open before a much smaller breaker protecting the wiring but dropping power to everything. A well coordinated system takes planning and the correct breakers and fuses to prevent this from happening.
It's very typical for the MCB to also be an RCD, and perhaps the device failed with a ground fault condition, causing an RCD trip rather than an excess current trip.
So are you going to get rid of that steamer, or what? If a device trips a circuit breaker everytime you use it, stop doing that! When a breaker trips frequently, that ages the breaker, that thing may not trip one day, and then, you'll trip instead! Or one of your family.
LH where are you? Reference to a fused switch indicates to me you are in the UK. I am guessing that a fused switch is a way of connecting a spur to a ring circuit. We don't use that in the US so I am sorta guessing that the current in the ring could be near capacity and so if one connects an additional significant load to a spur (which is below the rating of the spur fuse in the switch) one could trip the main breaker and not burn out the fuse in the connection to the spur.
If this is so, you should be able to operate the steamer on this circuit if you switch off enough other loads before connecting the steamer.