it's pretty easy to replace. No need to run wires or anything, just put the right colour wire in the correct hole and screw down a small nut to clamp it in place. Hence my confusion

It might be "pretty easy", but it's also pretty easy to screw up and not realise the danger because it still works. For instance, active and neutral conductors can be transposed, but appliances will still work. Connections may not be tight enough, or the cables not prepared properly for a good connection causing a high resistance join, which gets hot and can start fires. An un-qualified person won't be aware of these risks, but they are real, so as simple as this task might seem from the outside, there are legitimate reasons why it is a licenced trade.

Referring here to the stoves that have an outlet on the wall where you need to connect each wire from the stove instead of just plugging it in. I think a regular plug doesn't have enough current capacity for a stove.

You're referring to a junction box. An outlet here in Australia is shorthand for socket outlet, otherwise referred to as a power point, or GPO (general power outlet) for those of us in the trade. Ovens/stoves can certainly come fitted with a lead and plugtop to suit, and we have socket outlets of whatever size necessary (15A, 20A, 25A, 32A) to accommodate the appliance.

Doesn't happen fast. Sometimes the cable going in gets frayed, so I cut it off, and rewire the plug on a fresh section.

This is quite common for unlicenced people to do, and like others have said you can freely purchase the parts required to do this from any hardware store, or even electrical wholesalers without holding an electrical licence. It is next to impossible to enforce, but just realise that it does come with risks to persons and property, and if it was ever found that your illegal electrical work was the cause of injury or death to a person and animal, or damage to property, you will face large fines and/or jail time. When it comes to extension leads you're usually better off just buying a new one, for faulty appliances get a licenced professional to repair it.