What's the role of "blameless" in the Dhamma?

"Blame" means "blamed, censured or criticized" by the Buddha; similar to being condemned by the Biblical God Yahweh for your sins.

Buddhism is not an escape for ex-Christians from being blamed, judged and terrorized with threats of being sent to hell. MN 130 particularly shows the extreme horrific depictions of hell held to be believed & taught by the Buddha.

Also, there are many suttas where the Buddha appears to accept capital punishment is the inevitable outcome of certain bad kammas, including adultery (which many secular cultures such as in France appear to now hold to be OK, fun and acceptable; such as married men & even married women having mistresses & lovers). For example:

The blood you have shed when, arrested as thieves plundering villages, you had your heads cut off... when, arrested as highway thieves, you had your heads cut off... when, arrested as adulterers, you had your heads cut off: Long has this been greater than the water in the four great oceans.

SN 15.13

Then King Yama says, 'My good man, didn't you see among human beings kings — catching a thief, a criminal — having him tortured in many ways: flogging him with whips, beating him with canes, beating him with clubs; cutting off his hands, cutting off his feet, cut off his hands & feet; cutting off his ears, cutting off his nose, cutting off his ears & nose; subjecting him to the 'porridge pot,' the 'polished-shell shave,' the 'Rāhu's mouth,' the 'flaming garland,' the 'blazing hand,' the 'grass-duty (ascetic),' the 'bark-dress (ascetic),' the 'burning antelope,' the 'meat hooks,' the 'coin-gouging,' the 'lye pickling,' the 'pivot on a stake,' the 'rolled-up bed'; having him splashed with boiling oil, devoured by dogs, impaled alive on a stake; cutting off his head with a sword?

MN 130

Instead of legalistically searching for the meaning of words in dictionaries, it is best to read suttas for a contextual explanation:

Of course you are uncertain, Kalamas. Of course you are in doubt. When there are reasons for doubt, uncertainty is born. So in this case, Kalamas, don't go by reports, by legends, by traditions, by scripture, by logical conjecture, by inference, by analogies, by agreement through pondering views, by probability, or by the thought, 'This contemplative is our teacher.' When you know for yourselves that, 'These qualities are unskillful; these qualities are blameworthy; these qualities are criticized by the wise; these qualities, when adopted & carried out, lead to harm & to suffering' — then you should abandon them.

Kalama Sutta

Buddha: There are those families that were friends of the bhikkhu Channa, Sariputta, families that were his intimates, families that were blameworthy; but I do not say that to this extent he was blameworthy. Sariputta, when one lays down this body and clings to a new body, then I say one is blameworthy.

MN 144

The definition of "blameless" in the suttas is recursive--the definition uses "blameless" to describe "blameless".

AN4.135:5.1: And how is a person blameless?
AN4.135:5.2: It’s when a person does things by way of body, speech, and mind that are blameless.

To understand the definition, we have to consider the context. Blame is contextual. It is contextual because the context for blamelessness is the assembly. Living together in harmony requires blamelessness.

AN3.95:4.2: An assembly where the mendicants live in harmony, appreciating each other, without quarreling, blending like milk and water, and regarding each other with kindly eyes. This is called a harmonious assembly.

Assemblies are bound together by commonly held rules. Monastics will have the Vinaya. Yet even laity has rules. Even organized crime has rules. Those who break the rules in their society with body, speech or mind are not blameless.

Some rules assume the conceit, "I am". Importantly, Buddhist rules do not. For example, take the first three rules of MN8:

MN8:12.2: ‘Others will be cruel, but here we will not be cruel.’
MN8:12.3: ‘Others will kill living creatures, but here we will not kill living creatures.’
MN8:12.4: ‘Others will steal, but here we will not steal.’
...

Because Mara exploits loopholes with a passion, blamelessness requires utter vigilance. It also requires abandoning the conceit, "I am".

DN34:1.2.11: What one thing should be given up?
DN34:1.2.12: The conceit ‘I am’.

SN45.3:2.2: Good friends, companions, and associates are the whole of the spiritual life.

Without blamelessness, how could there be a spiritual life?

Most importand to relay on Samma-mana, right conceit, going after blameless. Without the notion of feeling rightly blameless, the path wouldn't develop and faultlessness is prerequisite for pathdevelopment Jhanas. Sure it's problematical to be accepted by on-house-holder and stand-maintainer as there way requires dusty blameful ways of which they are not willing to give up. In that way they start to develop a virtue out of incapacity, try to polish there faults with 1000 useless words of wrong conceit, maintaining there being.

By seeing blameworthy as blameless and viciversa, step by step they increase their bond downwardly, hoping that rebirth will not come along for those denying cause and effect anyway.

Most blameworthy those followers of the naked... most blameworthy their jeloose ways. Stingy in all regards incapable for Jhanas, Paths and fruits, burning off by defending decaying stand, house.

"We want a happiness that’s blameless, i.e., one that doesn’t harm anybody."

~ Thanissaro Bhikkhu "Brahmaviharas on the Path" https://www.dhammatalks.org/Archive/Writings/CrossIndexed/Published/Meditations8/151122_Brahmaviharas_on_the_Path.pdf

AN 4.62 defines the happiness of debtlessness and the happiness of blamelessness, but says that the happiness of blamelessness is worth a lot more than the happiness of debtlessness.

And what is the happiness of blamelessness? It’s when a noble disciple has blameless conduct by way of body, speech, and mind. When he reflects on this, he’s filled with pleasure and happiness. This is called ‘the happiness of blamelessness’. ......

Seeing clearly, a clever person knows both kinds of happiness: the other kind is not worth a sixteenth part of the happiness of blamelessness.”
AN 4.62

The following Dhammapada verses also talk about blamelessness.

If knowledgeable people praise him, having observed him day after day to be blameless in conduct, intelligent, endowed with discernment & virtue: like an ingot of gold — who's fit to find fault with him? Even devas praise him. Even by Brahmas he's praised.
Dhp 229 - 230

Blameless in conduct by way of body, speech and mind is related to virtue, and therefore also freedom from remorse.

Remorse is of course, what one feels, when one is blamed for his or her conduct. When one is blameless, one is remorseless. This leads to jhana according to the sutta below, and eventually to liberation. And that's how blameless conduct is related to the Dhamma.

Remorse can be considered self-blame.

"Skillful virtues have freedom from remorse as their purpose, Ananda, and freedom from remorse as their reward."

"Freedom from remorse has joy as its purpose, joy as its reward."

"Joy has rapture as its purpose, rapture as its reward."

(this goes on, till)

"Dispassion has knowledge & vision of release as its purpose, knowledge & vision of release as its reward.
AN 11.1

Having blameless conduct is also one of the four factors of stream entry.

And a noble disciple’s ethical conduct is loved by the noble ones, unbroken, impeccable, spotless, and unmarred, liberating, praised by sensible people, not mistaken, and leading to immersion. These are the four factors of stream-entry that they have.
AN 10.92