You can capture any sound/accent in Urdu. What I mean is you to take any word in any language and if you wanted to write it as-is in Urdu syntax, you can do so with 100% accuracy.
Well, except for all the tonal languages, such as Chinese (Urdu has no tonal system). And the various Caucasus languages such as Ubyhk, which has over 80 phonemes (Urdu has 48).
But your point is well-taken, which is that Urdu is a phonetically-rich language.
And as for the using persian script for urdu, without going into political stuff, I think it is a huge advantage - Persian script is not only used for Persian and Urdu, even Balochi, Daari (Afghanistan), and Pashto uses Persian script. Moreover, by virtue of knowing Persian script, it becomes immensely easy to pick up Arabic (at least reading it).
Then there are few other languages such as Sindhi (Pakistani Sindhi) which have a Persian-based script (few other characters on top of Persian which are specific to the language).
Which means, if you can read Urdu, with little effort, you can start reading Arabic/Balochi/Sindhi which means you can read literature from Pakistan all the way to Morocco.
The Persian alphabet seems to be lacking some vowels. How is that handled when writing Urdu?