Using a barrel jack with higher than rated current but lower voltage?
No, it doesn't work the way you think. At these levels, the voltage and current ratings are independent.
The current rating comes from the contact resistance and the power the actual contact (way smaller than the whole connector) can safely dissipate. This not only includes things like not melting the insulation, but also oxidation and other heat-related degradation over time.
The voltage rating is what the insulation is good for.
Multiplying the voltage and current ratings tells you the maximum power that can be pushed thru the connector. It has nothing to do with what the connector will dissipate. That's soley a function of the current and the contact resistance. In fact, it's exactly the square of the current times the contact resistance.
If you want to use the full output of your 5 V 10 A power supply, you have to use a connector rated to 10 A. It also has to be rated for 5 V, but it would actually be difficult to make a connector that isn't rated to at least 50-100 V, so the voltage spec is not a issue here.
You might ask the manufacturer of the power supply to tell you a specific mating connector it is intended for. Surely they know. Or they know there isn't one, but that consumers aren't going to think of that, and think a 10 A supply is "better" than a 5 A supply. The wire you show looks suspiciously thin for 10 A, at least with reasonable safety and loss.
Basically, your supply is like a 400 horsepower V8 for a Honda Civic. You can't just plug in either one without dealing with other issues.
Yes, this is really an issue.
Current rating is mostly based on the power dissipated by the contact resistance.
P = R * i2
So, the heat generated on a connector is about the current, not voltage.
The voltage and current ratings on connectors are independent. The voltage rating depends on insulation strength, while the current rating depends on contact resistance and construction.
You may be able to use a 5 Amp connector for somewhat higher currents, but it may overheat. I don't recommend it.