Does omitting the 周囲 change the effect of its use in sentences or is it more similar to an idiomatic set phrase?
Yes. Since 取り囲む is a verb, and it is a transitive verb ( you can imagine easily, I think ), so the object is necessary.
Now, onto 周囲
Please take a look at here
http://okjiten.jp/kanji676.html the word 周
Here is how the Kanji 周 was born. You can see the 4 blocks. And you can see also
How was the Kanji made?
This is 指示文字 ( I do not know the corresponding English, sorry ( But here http://okjiten.jp/kanji1.html we can learn the 4 patters how Kanji was created ). From the pictograph that looks like a rectangular container or other tools sculptures are decorated outside, the Kanji 周, denoting "all over, far spreading" was made.
the word 囲
This is 形成文字. From an ideograph denoting "a line surrounding an object" ( enclosure )
and another expression "footage each pointing different directions" the Kanji 囲む
denoting "to build a fence" was made.
So, 周囲 is ”far spreading fence ( field ).”
So, when we say ビルの周囲, we mean "the vicinity of the building" due to the derivation
from above, like a fence circling the building.
When you say, 周囲を取り囲む, sure, as you say
it seems to describe the action of crowding around a particular thing or object (people crowding around someone,
when people crowd, it becomes a "mass", right? So
ex:（人々の）周囲を取り囲む surrounding people ( of some kind ).
Have a nice Sunday.