You could run on 5V with a boost voltage converter to 12V. For extra credit, you could run the USB-PD off 5V, negotiate 12V and only then switch it to the load.
That's not to say that a boost converter doesn't have value, but it still leaves a gap where there could be confusion.
The confusion or complexity even multiplies if the device has additional USB-C for data transfer. In that case, you either have to mark one port as being the "power in" port, or you have to support power in and data out on all the ports, which gets complicated and expensive.
It would be a great move by the USB IF to think through this sort of thing more carefully. Right now the USB-c connector is so overloaded in terms of power, display modes, thunderbolt, speeds, etc. that it's very hard to predict whether two USB-c devices will connect and at what power or speed and with what capabilities. For power, it would be helpful to require supplies to have a standardized status LED so that e.g. green means that the supply is providing the highest power allowed by the device (not the supply), yellow means there's been a compromise, and red suggests an error condition.
And yeah you're supposed to negotiate before pulling 2.4 amps at 5v but that's not usually a big deal in practice. Especially when you're actually supposed to stick to 100mA at first, but who does that.
A diagnostic LED sounds nice but given how most cables don't even have a speed printed on them good luck at something more invasive.
I will say that thunderbolt support isn't often an issue beyond the basic speed rating, and should be even less of one since USB4. And that power ratings are pretty simple, 60W or moreW. I really don't think the overloading of many different types of feature is a big deal, I think the single feature of unknown speed is the big issue-causer.
But it's a bit more involved than just replacing a barrel jack with a USB-C port, and would require some design considerations early on