I'm torn here.
Yes, some people use "where's the evidence" as a conversational gambit to try to shut down discussion they don't like. (And if you provide evidence, they may say "that's only one source, got any others?") And if they're being dishonest in asking, there's no point supplying the evidence they request; it is useless to try to have a conversation with those who will not listen.
On the other hand... when you make a claim, the burden of evidence is actually on you, not the other person. And if you say "you have access to the same search engines as everyone else", well, that's true. On the other hand, one person writes a post, and ten people read it, or a hundred people, or a thousand. Making the thousand do the searching, instead of having the one writer do it, is really inefficient.
This leaves you at "do the work of providing the evidence, but don't feed the dishonest trolls", which is... well, at best it's not very actionable advice.
I told you I was torn...