12v DC brushless PWM fan burnt out via Arduino Uno
With just the Uno connected, the voltage from the supply is around 16V!

Looking on the internet such dimension are definitely not typical nowadays regulated SMPS.

Figure 1. 3COM 3c16741a from an online store.

I predict that your power supply is a typical unregulated power supplies which only guaranteed to deliver 12V only at specific current load. This is a typical schematic of those.

Figure 2. Typical unregulated old-fashioned “linear” transformer-based wall wart. Image Source
And here is the typical Load Regulation graph. It's analogues for 12V 1A in your case. For the unregulated power supply, when there is no load or little load the voltage would be very high. As the load increases the voltage will drop proportionally. On the other side, any typical regulated power supply would maintain the regulation of the output voltage if minimum supply load current is sufficient such as 10mA of LM7812 Linear regulator. And then temporarily shutdown when over-current or over-temperature occurs.

Figure 3. Load regulation: Unregulated vs Regulated. Image Source

Conclusion at the moment
Moreover considering

Briefly turning the supply on made the fan turn very fast, as expected.

I highly suspected, you're overvolting the fan by using unregulated power supply.