"After all, for a fuse to work, it needs to melt metal inside it."
Can I just add a comment here for those who are not as understanding as I think it might , in some circumstances, cause confusion?
The "work" of the fuse is to pass current and all conductors heat up when passing current.
By heating up thy might get very hot or a bit hot or slightly warm or not noticeably warm , it all depends on the amount of current drawn and a few other factors.
When a fuse does blow it does get "very hot" in order to melt it and break the circuit flow.
The fact that you have noticed that the fuse is hot does suggest that it is excessively hot and this could be because of overcurrent or by a loose connection, it could even be because the plug or socket is cheap/inferior rubbish.
Sorry for mentioning this and you might think I am deliberately being pedantic, I`m not trying to be though I just saw a couple of things said that could be misconstrued by a few. We all say things at times that could be misconstrued, I certainly do!
Example "My Burglar Alarm went off!", not it didn`t it went "on" or it "activated". That`s why we use words like Set/Unset, Open/Close, Activated/Deactivated etc etc in the hope of avoiding confusion. I have known customers to use the term "Went Off" to explain the lights/LEDs going unlit because mains power was interrupted and battery was depleted rather than meaning that a sounder activated.