His purpose wasn't specifically to show mercy. It was to use her to illustrate we've all sinned and judgment and punishment should reflect this fact. It's to show everyone should be judged accordingly rather than black and white, pretending there are two groups: sinners and innocence. There is only one group, sinners.
Let he who is without sin cast the first stone. Judge not least ye be judged, contrary to popular myth, doesn't mean don't judge. It means judge accordingly as you would wish to be judged else you may be hoisted by your own petard.
If you're mindlessly killing all sinners, it means you too qualify for death. As such, mercy, law, and reason is called for, for there to be justice. Mob violence isn't necessarily justice. It wasn't specifically for mercy for her. It was mercy for all of us.
To add to this, "Go and sin no more" is only merciful compared to death if someone's willing to pick up that metaphorical cross, bear it, and change their ways.
It's not some megachurch "All is forgiven, keep on sinning" pablum. It's akin to telling a morbidly obese person "Go and hamplanet no more". It requires them permanently and drastically change their diet. They'll have to give up most of their friends who encouraged horking down cheesecake. They'll have to lay down dramatic boundaries with their family who raised them to live such a destructive lifestyle. They'll have to abandon their couch potato hobbies and adopt new ones which are literally and figuratively painful. If they join a volleyball club they'll be picked last and feel the shame of being a hindrance to their fellows.
Telling a ho "go and sin no more" is just as much of an ask in exchange for forgiveness.
Nicely put.
Excellent points.
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