Weaponized anti-science attacks people with statements that sound credible at their level, and require higher level to understand the deception. This does not happen accidentally; it is constructed so on purpose. So the person with elementary-school knowledge reads the statement that seems to make sense from the elementary-school perspective (it does not contradict anything taught at elementary school), but a high-school students could see why it is completely wrong. So the person can only conclude: "as far as I can understand, it is correct... and there is this one guy saying that it is wrong, but his arguments I can't understand". Similarly, you target people with high-school knowledge with statements involving undergraduate-level deception, etc.
This is for me the most frustrating part of debating smart people who believe pseudoscience -- as far as they know (in my bubble that typically means undergraduate level of math, plus high-school level of everything else), the science often checks out. Someone who graduated in given area will be completely frustrated by hearing the "proof", because it contradicts many known and well-tested things... but to convince the person who is ignorant about it all (but is completely willing to trust his own Dunning-Kruger syndrome and Nassim Taleb's twitter over thousands of peer-reviewed papers), you need to hunt for the subtle error in the mathematical proof.
Indeed. However, basic media literacy training could help people to differentiate between quacks (Alex Jones, un-attributed opinions in social media share pics not grounded in any scientific facts), actual experts but not in a relevant field to a debate (e.g. a PhD in art history or psychology in a debate about the infectiousness of coronavirus), and actual experts (e.g. Christian Drosten, one of the discoverers of the first SARS Coronavirus).
A bit of common sense in media entities such as newspapers in TV stations - such as not giving quacks and non-relevant experts airtime and having scientific expertise on-call - would do the rest to further civilized debate.
There is another point when "skepticism" as a principle fails, and that is when it is being weaponized by political and/or financial interests. Climate change is the most obvious example with oil companies funding all sorts of quackery, derailments and misinformation (such as BP did by inventing the "individual carbon footprint" in the 70s, see https://mashable.com/feature/carbon-footprint-pr-campaign-sh...), but also the tobacco and alcohol industry - and in the last two years, COVID deniers who branded themselves "skepticists" only to renounce the most basic bits of societal solidarity such as wearing masks and getting a recommended vaccination.
Skepticism should always, always come with at least some provable (!) claims - for example, with COVID vaccines, the issues regarding AstraZeneca and blood clots. These claims can then be investigated and validated or disproven. The problem is that modern and/or "alternative" media simply takes skepticist claims and reports them as gospel either because they don't have journalists with a science background any more or (as is the case with OAN/Newsmax/Fox/Murdoch/Axel Springer) they follow their own/their owners' agenda.
The root cause at the bottom of weaponized skepticism is the truly horrible state of teaching and the attitude towards science in schools in most Western countries. "Creationism" should not be seen on any curriculum worth its name, the fact that students get ruthlessly bullied as nerds/know-it-alls/autists/<insert whatever label you were bullied with here> without consequences and that such behavior is culturally accepted (e.g. in "teenage movies" and sitcoms, where "nerds" are routinely portrayed in absurd cliches), the total lack of media education (aka what bias is, how it works, and what the pitfalls of modern mass media like clickbait, deceptive ads masked as journalism etc. are)... our societies are rotten from the core.