Now the US/UK supply voltage shouldn't matter, correct?
Correct.
The internal PS in the device just wants to see 9-12V? Do I have that right or am I missing something?
You have that right - but there are 4 factors you need to get right.
Could I just use a typical US wall wort PS?
You have that right - but there are 4 factors you need to get right.
Or, will I run into polarity or cycle issues?
The input frequency (50Hz/60Hz) will not matter at all, but you
have identified one of the 4 factors you need to get right.
They are:
1. Voltage. There is the obvious need to match a 9V requirement with a 9V supply - but you also need to be a little careful here. Some supplies are not as well regulated as others and these can have a higher (sometimes significantly higher) voltage when the load current is low. Such is the case with many (not all) transformer based supplies (these are almost always heavier and/or larger than switchmode supplies). The danger here is that the voltage presented to the device is too high for it to handle and damage could result. This problem is minimised when the voltage and current rating of the supply matches that of the device as the device will load down the supply. This is pretty common for transformer supplies.
As a rule, switchmode supplies do not have this problem. You can identify them by being lighter and smaller than transformer types - but if you're not sure about this, then check the mains input voltage on the supply. If it says something like 90V-250V, then it will be a switchmode supply. As mentioned above, there is a caveat with switchmode supplies, especially used with audio devices, and the name gives you a big hint... Switchmode supplies work through switching a current on and off at a frequency much higher than the mains. Switching currents can cause EMI - both radiated and/or conducted - which can sometimes creep into the audio path. Much gnashing of teeth will ensue if this happens.
2. Current. Obviously, if a device needs a particular amount of current, then the supply
must be capable of supplying
at least this amount. Aside from the previously stated caveat regarding poorly regulated supplies, there is no problem with it being
capable of providing more current than the device requires - the device will just take what it needs.
3. Connector. This is a simple mechanical matching of the plug and socket between the supply and the device. Your typical barrel connector has 3 main dimensions - internal diameter, outside diameter and length. Get these right and you'll be fine.
4. Polarity. Centre positive or centre negative? Getting that wrong could ruin your day.