Is the sentence wrong without the 都？
Why the sentence is only correct / better with the 都？
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都 is an adverb for 'All' ; 'without exception'
"每个人带着甜蜜的笑容。" is not grammatically wrong, However, having the adverb 都 before the verb 带着 (carry), add emphasis to the fact- 'All of them/ without exception'
'人人带着愉快的笑容' = 'everyone carry pleasant smile on their faces'
'人人(都)带着愉快的笑容' = 'everyone, (without exception), carry pleasant smile on their faces'
It would be easier to see the difference if you replace 每个人(each person) or 人人 (everyone) with a term that hasn't already indicated the meaning of 'all'
'外面的人们带着愉快的笑容' = 'the people outside, carry pleasant smile' (we don't know it is some of them or all of them)
'外面的人们都带着愉快的笑容 = the people outside, all carry pleasant smile' (now we know all of them are smiling)
甜蜜的笑容 = 'sweet smile' , more suitable for describing smile of beautiful or cute girl
I agree with Tang Ho, 都(全部）is used for emphasis, 'all, everybody'. I don't agree that it is an adverb. I would class 都 as a pronoun in this usage.
都还好吧？ All good? / Everything OK?
We use 'all' in the same way in English and other languages:
We want some soup first.
We all want some soup first.
Do you need help?
Do you all need help?
They refuse to leave.
They all refuse to leave.
我什么东西都吃。(什么 could indicate a question. ‘我什么东西吃？’都 changes that to 'anything, all')
I eat anything.
I eat anything.
*我吃什么东西都。（This word order is not used. No idea why.)
I eat anything