The similar PPA (Pay Per Article) might work using LN, but it still not a convenient option for visitors even the traditional fiat method, look at its popularity.
Even if it's as cheap as 1 satoshi, the mainstream will think twice before considering the hassle.
I suppose it would depend on just what kind of visitor we're talking about, and just what kind of service they're trying to access on the site. I've seen some sites implement what it would call a PPA tool as well now, requiring visitors not only to complete a captcha but to complete a set minimum of hashes on their CPU power - takes about a half a minute to complete. A hassle most people wouldn't put themselves through, as you say, but it's a service I somewhat require almost daily and have so far been willing to complete. I imagine the method has worked quite well for the site, I don't see why LN micropayments wouldn't too.
Come to think of it, there's so much hassle I put up with when signing in almost daily to several services I use. If there were a way to automate all this, reducing time spent, I wouldn't mind making the regular (small) payments.