voltage-min-design:  11.58 V

This has nothing to do with USB-C, this is the minimum design voltage of your lithium ion battery pack. In this case, you have a 4-cell pack, and if the cells drop below 2.895V that means they're physically f*cked and HP would like to sell you a new battery. (Sometimes that can be fixed by trickle charging, depending on how badly f*cked the battery is.)

If your laptop's USB-C circuitry were built for it, you could charge it from 5V. (Slowly, of course.) It's not even that much of a stretch given laptops are built with "NVDC"¹ power systems, and any charger input goes into a buck-boost voltage regulator anyway.

¹ google "NVDC power", e.g. https://www.monolithicpower.com/en/learning/resources/batter... (scroll down to it)