Tanada, using those screw-in circuit breakers that work as you described is illegal and extremely dangerous in every jurisdiction in the USA/Canada as they are NOT COMPLIANT with any National Electrical Code published since the 1970s.
BTW, what you described are "auto reset" thermal breakers, which are only usable in the USA in vehicular electrical systems, and are now banned in residences. (They never were legal in Canada.) They are not even a good idea in automobiles and trucks. When an overcurrent occurs, power needs to remain off until the fault is corrected.
The devices you showed a picture of are "manual reset" fuse replacements which trip after a prolonged overcurrent which as you noted, cooks the wires. Those breakers are reset manually by the black button in the center. Those that you showed pictures of are illegal in almost all jurisdictions, which require an Underwriters Laboratory and/or Canadian Safety Agency safety rating and marking - those are made in China and lack the UL/CSA markings.
The generic problem with fuse replacements is that older wiring is not the same as modern wiring which has insulation designed to tolerate the temporary overcurrent and to preserve insulating properties at higher temperatures. For example the Romex wire commonly used for residential wiring in wood-framed buildings:
This wire has a temperature rating of 194 degrees F. That sounds like a lot, but it does not take a prolonged overcurrent to melt the wire insulation in an attic space that is already at 140 degrees F on a hot Summer day. The interior insulation is wrapped with brown Kraft paper which has been treated with Boric Acid or other fire retardants. Contrast these wires to the 70-100 year old wiring that one would find in a residence old enough to have screw-in fuses, which would likely have cotton-wrapped wires in a woven cotton jacket impregnated with now-100-year old oil-based enamel. Otherwise known as kindling.
Which is why I am a believer in metallic conduits with metallic junction boxes that contain fire, when I spec my new house for construction. Except outside where for trench burial I favor plastic conduits that will not conduct massive amounts of electricity into the residence itself from nearby lightning strikes.
The painful truth: if you are experiencing an excessive amount of fuse replacements, you need to have an Electrician rewire all or a portion of your house, and eliminate the overloaded circuits. A total re-wire with modern wiring and a modern circuit breaker panel is the ideal corrective action. If you cannot afford to do that, you need to stick with the original fast-acting fuses.
Sorry to go all "EE" on you, but professional ethics will not allow me to pass when somebody is giving bad advice that potentially could harm others when followed.