The problem with circuit breakers is they require a huge overload to open instantly, and will allow a slight overload from several seconds to a whole minute before opening!
It is the same with fuses, depending on the specific fuse type.
In general the time to blow rating on a fuse will include a rating time to blow at 100 percent overload, that is twice the rated current. This is usually at least 60 seconds or more.
That is a 5 ampere rated fuse will handle up to 10 amperes for at least 60 seconds.
Clearing (Total operating) Time.
The total time between the beginning of the over current and the final opening of the circuit at system voltage. Clearing time is the total of the melting time and the arcing time.
Of course this depends on the type of fuse.
Of course one can choose one of the many varieties of fast acting fuses. However, these fuses then would have to be sized to handle any brief overloads that might occur, such as turn on surge current, which might be several times the normal operating current.
As a visual example of this, think about the operation of the DBT (DIM BULB TESTER) and how the light bulb may flash to full brilliance when a piece of gear is first turned on, then settle down to a very dim glow on a correctly working piece of gear.
If a fuse is not sized correctly it may fail prematurely when subjected to repeated short duration overloads.
Fuses that seem to fail prematurely may indicate and issue with a piece of gear or an improperly selected fuse, either as to type or rating or its installation. Even the type and condition of the fuse holder may have a measurable affect on the performance of fuse.
The selection of a circuit breaker or fuse for a particular application from a design perspective is not as simple as it might seem at first glance.
In general, as has been stated before, a circuit breaker will still have some I^2*R loses, although there are ways to reduce this in more sophisticated circuit breakers.
Current transformers and magnetic field sensing devices are couple of methods, but they may become a little pricey for general use.
Of course there are the myriad audiophile fuses that are available, however that is
another can of worms.
BTW, a sudden measured resistance, thanks to Mister Ohm, is actually current. Remember, resistance is a measured voltage drop at a specific current. You can not measure resistance with out a voltage drop.
One could endeavor to reduce this resistance, however a normal fast acting 5 ampere fuse may have a Nominal Resistance of about 0.02 Ohms, rather small in the overall scheme of things.
The key element here is safety. If one were to use an electronic means to measure the voltage drop across a specific very low value resistance and use this signal to trip a circuit contactor, if you have a failure in your
fuse electronics so to speak, then you may have a safety issue.
There are already some very sophisticated methods using electronic circuits for circuit protection protection, but for everyday use it is hard to beat a properly selected fuse.
Well, shielding can keep the hum in (conserve energy! 
