That PEW survey is generally trusted in academic circles, although its implications are naturally debated. The high favorability rates for Sharia law do not necessarily point to sympathy for specific punishments for specific crimes, and some high rates in favor of say, death for apostasy does not mean that every person will kill an apostate when they see them. However, the positive associations that the term ‘Sharia law’ creates among many people in Muslim countries is meaningful to note in itself, and there’s a lot to be said about it. Much of the criticism that we can make from it actually can be directed at modern governments in Islamic states, especially those that claim that their regimes are compatible with ‘Islam,’ and the international structures that support them. Other criticisms involve the assumed sacredness, goodness and authority of the primary sources and to an extent those recognized to expound on them; and of course there are also history-based critiques of the widely recognized theoretical legitimacy - if not the local applicability - of disagreeable practices, like stoning or slavery or the enforcement of patriarchal norms.
In any case, these discussions should always be done with empathy shown to the people involved, and an awareness of intersectionality issues.
Certainly, many Westerners - liberals and conservatives - do Muslims no favors when they define as peace-loving anyone who denounces the terrorist group of the day and turn a blind eye to violence - immediate, structural, cultural - toward or within Islamic communities.
What most people in my experience fail to grasp is that Islam, like other religions, exerts influence and is influenced. Local customs gain local approval or local opposition within an Islamic framework. What this means is that you usually cannot say that Islam in one place says very much about Islam in another place. What it also means is that you usually cannot say that some local custom in an Islamic community is ‘unrelated’ to Islam. In most cases, the relationship is not “Islam as a whole says to do this” but rather “the community has, at this time, reconciled these practices with their religious identity in some form.” Mealy-mouthed, yes. But more accurate and helpful. For better or for worse, the only thing that really unites Muslims around the world is the issues and sources upon which they disagree.