If you look on my house (USA) where the power comes into the house you see three wires. Two insulated and one bare cable. They connect to the house via crimps.

One question:
Are there any advantages of a solder joint vs a crimp wire joint?

Each has its best application. In the case of high temperatures crimps tend to perform better than solder. High temp solder is better than standard solder for high currents and high temperatures, but in the case of your cooktop the crimps are the way to go. More reliable than a meltable joint. Braising has been mentioned. That is better than solder and PROBABLY better than crimps, but it takes more work and skill to produce a good braised joint. From a manufacturing standpoint crimping would be the fastest way to manufacture. And the reliability is sufficient for the life of the component. As you have seen, the element failed midway in its circuit, not at the crimp. But I would also suppose that the crimped joint is also made by preparing the element for such mechanical connection.

 

85% silver solder has a melting point of ~740C. But, getting the solder to "wet" the element could prove very difficult. "Hard" silver solder used in jewelry making is the sort that might work. You will need the right flux, and mechanical preparation of the parts to receive solder (abrasives) will probably be necessary. A small torch of the jewelry making sort, using butane, is probably your best bet.

There are plenty of resources about hard soldering and the process online.

Usually, you'd weld or make mechanical connections but that seems impractical.

Good luck.

"flogging a dead un" is an adage that springs to mind...…………..

 

Just replaced the piece. It's working fine. The piece uses mechanical joints originally, crimp wire terminals. Indeed, soldering didn't work at all, the cable didn't catch any solder at all, it was like solder-phobic.

One question:
Are there any advantages of a solder joint vs a crimp wire joint?

First things comes to my mind is size: crimp wire terminals I guess only really work when cables are think. Tiny electronic cables such as audio cables or smaller can't be managed with crimp terminals.

There are a HUGE number of different kinds of crimp terminals, including types that are approved for use in military aircraft. And some of those do use rather thin wire. But the terminals we see in appliances and cars are the really cheap ones that are not nearly so reliable. So for the average person with decent soldering skills, a soldered connection is better. But for some folks who can't solder well, a crimp may be better.
The advantage of a good soldered connection is that every strand is connected, and that the connection is completely sealed against corrosion by the solder. The secondary benefit is that the connector can be re-used if the wire fails. That can be a big advantage if you are trying to fix one wire in a multi-pin connector and you do not have another pin.

 

Something that hasn't been mentioned - ultrasonic welding. That chit is pretty darn good and reliable. However, owning an ultrasonic welder may be a bit pricy. Just thought I'd throw that out there.