Did Natalee's family try to portray a "perfect" image of Natalie? Did they think that only a non-drinking, non partying young lady would be one that people would care enough about to look for?
I don't think it's fair to assume that Beth deliberately concealed Natalee's partying activities on Aruba. Parents don't always know everything their children are doing. According to Natalee's Mt. Brook classmates, Natalee was not typically a heavy drinker, and I doubt that when Beth first arrived in Aruba, she had any idea how much Natalee had been drinking there. I wouldn't be surprised if she didn't find out for weeks, or even months.
Furthermore, with respect to all exaggeration of Natalee's virtues by her family and friends, whether accidental or deliberate, it's important to remember the context. Natalee disappeared under circumstances- leaving a bar with male strangers- which would inevitably cause many people to immediately form certain negative assumptions about her character. With people asking "What kind of girl would get in a car with three boys she didn't know", it was only natural that her family and friends would feel the need to defend her. If they were somewhat overzealous in doing so, that was a mistake, but not one they deserve a lot of criticism for.
Also, if they were concerned that Natalee's image would effect how much people cared about the search for her, they were, unfortunately, correct. Compare Natalee's case with that of 17 year-old Taylor Behl. Taylor was also a telegenic "missing white woman", indeed one who was more attractive than Natalee, and who went missing at about the same time. But her disappearance was receiving less national media coverage at 3 weeks, than Natalee's did at 3 months.
I think part of the reason for the disparity is that Taylor had been involved in a previous sexual relationship with her then-suspected (and later admitted) killer. Natalee's story was perceived by many parents as a cautionary tale: If an archetypal "good girl", a responsible, straight-A student with no history of promiscuity or drug abuse, could experience a temporary lapse of judgement and pay for it with her life, then their child could as well. But Taylor's case didn't resonate the same way, because parents could tell themselves that their daughter wouldn't have a relationship with a weird character twice her age. The reality is that a female victim's behavior and sexual history are factors in how sympathetically the public views her, and how much interest they have in what happened to her.
Was ALE's investigation misdirected by Beth's idealized description of Natalee?
I see no evidence to support that idea. On the contrary, it seems ALE was initially operating under the assumption that Natalee had voluntarily gone off somewhere to continue partying, as other tourists had supposedly done in the past. When Dave Holloway first arrived on the island 48 hours after Natalee was last seen, the first two police stations he visited were not even aware of her disappearance, and at Noord station Detective Jacobs told him she was probably out partying and might show up at Carlos and Charlies' for Ladies Night. So clearly, Beth's description of Natalee as a responsible non-partier, hadn't made much of an impression on ALE.
In fact, even though the Natalee portrayed by Beth was idealized, ALE would actually have been much better off if they had completely believed in Beth's Natalee (because, however inaccurate, the idealized version was closer to the truth than ALE's assumption that Natalee was so irresponsible, she was capable of missing her group flight home and going off somewhere for days without telling anyone where she was). Hence, as unreliable as Beth has been, it is ironically, nonetheless true that ALE's initial and biggest mistake wasn't that they listened to Beth, but rather that they didn't. If they had, they would have taken the case seriously right away, and not wasted the crucial first 48-72 hours.