The speaker wants to indicate that they are conscious that "giving the middle finger to the society" is a somewhat cliched figure of speech. Adding "proverbial" to "middle finger" is just an arch way to acknowledge that it's a somewhat shopworn expression. "Proverbial" here means something closer to "hackneyed", rather than that there is a literal proverb about giving people the middle finger. The association comes from the fact that many proverbs become trite through overuse.

Well maybe it's just me, but I got the feeling the OP has some religious concerns involved in the question. if and only if that's the case I would like you to consider the fact that some people just don't have religious attachments or restrictions attached to their minds. So some words that may be used with great care and respect by some, simply don't have the same or any value to others and may be used carelessly. Probably a person that doesn't have religious attachments won't even think that someone may attribute a sacred meaning to some words, because to them there's nothing sacred in the first place. Again only if that's the case I would like you to consider that as every religious belief should be respected, non-religious mindset should be respected equally. So try to separate the association between the religious sense of the "proverbial" word and its secular meaning, and maybe you have a better understanding of this expression.

It is to be dramatic

One way to say it would be:

After careful analysis and consideration of all the various factors involved we have agreed to come to the general conclusion that we do not think there is reasonable merit to the idea.

eh

It reads better (to some) and has more effect (again to some) as

We gave it the proverbial middle finger !!!

because of the terseness and also the image and subsequent thoughts it brings up - ('f' you, 'screw you' is the language behind middle finger often.

The appropriateness of it will depend on the social scene and settings.
Maybe ok for a night out party. less so for a work meeting ;)

You could certainly replace proverbial in the original sentence with metaphorical or figurative and preserve most of the meaning.

Use of "proverbial" here does two things though - as well as signifying the sentence isn't meant literally, it's also calling out the fact that "giving someone the (middle) finger" is a colloquial phrase that is often used non-literally like this.

Isn't it more suitable to say:

We've given the figurative/metaphoric middle finger to the society?

That's exactly the meaning of the phrase, yes. Typically 'proverbial' in this sense is used to refer to an actual well-known proverb or idiom. The M-W definition

1: of, relating to, or resembling a proverb
2: that has become a proverb or byword : commonly spoken of

Gives a great example: 'the proverbial smoking gun'. This doesn't refer to any actual proverb, rather, the well-known idiom 'smoking gun'.

The usage in your example is odd in that they use 'proverbial' to refer neither to a specific proverb nor to a common idiomatic expression. I'd almost go so far as to call it wrong.

A proverb is an idiom, metaphor, or witty saying that is well known and in general use.

On the other hand, The Book of Proverbs is a collection of common sense wisdom and advice located in the Bible.

The middle finger in the US and some other countries is a well known and commonly used (amongst some) obscene gesture meant as a pro-verb meant to silently replace the verb-like, action phrase, “GO FORNICATE YOURSELF!!” It is synonymous to placing the thumb between the middle and the ring finger in some countries. Or, giving the peace sign (index and middle finger extended only) with the back of the hand facing the intended recipient. Or, thrusting the fist upward and stopping the motion with the opposite hand against the inside of the elbow.

Since the middle finger is well known and in general use, it is proverbial.

In this example, the choice of "proverbial" seems to employ irony - there are no proverbs (to my knowledge) that refer to the rude hand gesture in question, and it gives the quote a dark humor. "Figurative" or "metaphorical" would be a more literal choice, but would remove the ironic flavor.

Per M-W, "proverbial" doesn't necessarily need to be used to refer to a term's use in a proverb; it can also be used to refer to use in an idiom.

proverbial = goes beyond its first meaning

Merriam Webster:

Definition of proverbial 1: of, relating to, or resembling a proverb 2: that has become a proverb or byword : commonly spoken of
the proverbial smoking gun

aka well-known or familiar, too.

No, there is no specific "proverb" associated with the middle finger. However, the middle finger gesture as an insult is well known. This is expressed as: to give someone the finger.