I fully understand how they were their own worst enemies as far as public perception of them was concerned. I didn't see and hear a lot of what went on after the murder (TV, tabloid, etc), but there are parallels with the Chamberlain case here. They too were frequently interviewed on TV and parents, especially the mother, came across as cold and calculating. People could not reconcile her behaviour with someone whose baby had been taken and most likely eaten by a dingo. One TV interview especially (some time after) probably clinched it, with LC showing no emotion as she described in detail how she thought the dingo had killed and dismembered her baby.
The difference between the two was the Rs had money, the Cs had religion (he was a Pastor) and in people's minds, they similarly thought themselves immune from prosecution. The depth of feeling against LC both in the public's and Police's perception gave them a 'let's get 'em' attitude. The investigation and the forensic evidence were similarly suspect. The only difference here is that the body was never found but despite this LC was convicted of murder and spent quite a few years in jail before being cleared and freed (with a nice little compensation pay out as well).
Regardless, there would be a large percentage of people who were around when all this happened who still believe LDI. Her years in jail did not soften her image.
So yes, I understand how an image presented can sway both the public but more importantly the Police and direct their investigations right from the beginning. That's what I think the main problem was here with the Rs and if as HOTYH suggested, there had been an RDI and an IDI team, each independent of the other, the evidence we have to work with would likely have been very different.
Apparently, they heeded the advice of their lawyers, too, who may not always be as devoted to the management of public image and perception as other professionals.
JonBenet was cute, blond and wore make-up during a number of public performances; She was dressed in expensive, custom tailored outfits, appearing to be older or trying to appear older; She danced and sang before audiences, all at six years of age. Apparently, some people interpreted these gigs as ostentatious displays of a mother's gloating, selfish ego, even to the point of exploiting a voyeuristic, sexual, mystique of some kind.
A certain bias/envy/resentment/suspiciousness against/of the wealthy, especially the newly rich, played a part in the country's love to hate this woman, too, I think. Again, make Joni a girl scout, dressed in a hand-me-down uniform, the child of working class parents, struggling to make ends meet, and they would have faced less public outrage.