Sentence 2 is not OK. Removing "while" and "are" transforms the first clause into a fragment, incorrectly placed ahead of the next clause. "Most politicians" is the subject, but here is no verb associated with it.
Put into a more natural order, sentence 1 is:
- This candidate stresses that improving the lives of the citizens is a matter of import, while most politicians are downplaying the dire economic situations the people are facing.
Still not good enough, sentence 1 can be transformed into:
This candidate stresses that improving the lives of the citizens is a matter of import, and most politicians are downplaying the dire economic situations the people are facing.
but a good part of the meaning is lost.
Most politicians downplaying the dire economic situations the people are facing ...
can be correctly used in a sentence like:
Most politicians downplaying the dire economic situations the people are facing are now in vacation.
In this last example, the same fragment is an absolute phrase.
In the case of an absolute phrase, the absolute phrase is strongly related in meaning to the rest of the sentence. In the original example, "most politicians" and "this candidate" are actually presented as conflicting, not as a unit.
To better understand the absolute phrases, you can study the following examples (from here).
Marshall held onto the ball, his fingers squeezing it tightly.
(Marshall ... his fingers)
I will be back tomorrow, weather permitting.
(as in: if the weather of tomorrow permits)
Her voice floating over the crowd, Maria awed everyone with her natural singing ability.
(Her voice ... Maria)
Sam could be heard all the way down the hall, his loud laughter carrying across the school.
(Sam ... his laughter)
Her fur matted and her legs shaking, the lost puppy finally found her way home.
(Her fur ... the lost puppy)
And also the example you provided in the comment:
The putrid smell of the carrion having attracted a group of vultures, we decided to spend the day taking photographs of the birds.
(vultures ... the birds)