Fortunately (or not as the case may be) we don't have to guess as to whether or not culling works, we know for a fact that it does. Now, to what degree it helps, and also to what degree we are helping nature find a "balance" maybe we have a little less data on.
I've been involved in the whole lionfish thing pretty much since it started here on Curacao. I've been to lots of seminars, and had more than a few beers with more than a few marine biologists (well except for Fadilah Ali who prefers ice cream to a cold beer). Those little suckers have been around long enough now that we have actually been able to compile bio mass data over the course of years.
We know that culling will not "solve" the problem. IE there will always be lionfish around. They are to prolific and can survive way too deep and for that matter way too long without food. What we do know about culling is that in areas where it is done on a regular basis the bio mass of lionfish is down. This has been studied for sure on Bonaire and Curacao, and I imagine other places as well. We also know that the areas where culling is taking place, the shallow water reef systems, is an area of massive importance in terms of the percentage of life it sustains as well as its importance as a recruitment and hatchery area (photic or euphotic zones and all that). So keeping the LF problem in check in areas where most divers can dive is actually a pretty substantive impact.
Now as to nature finding a "balance" one of the things we have to remember is that we are part of nature. Mankind and its hunting tendencies can indeed have an impact, and a profound one at that. We frankly are awesomesauce at killing sh$% when we want. We also happen to know from scientific studies what happens in areas that aren't culled (70-90% of all biomass eliminated by lionfish in less than a year). So sure nature can find a balance without culling, it will look like a barren wasteland underwater, but it will indeed balance out.
As for natural predators, everything I have read and heard suggests that this will take centuries at least. Sure there is anecdotal, and even video evidence of it, but it is very rare. On the other hand thousands of fish are caught and disected every day, and we aren't pulling up groupers left and right and finding LF in their bellies. Heck the only fish that we find LF inside of on a regular basis is, you guessed it, other lionfish! As for people that go out and feed culled LF to predators, well there are a lot of words to describe that practice, but my grandmother told me not to use those words in a public forum.
I've met the people you (OP) are talking about that are all anti culling "nature-will-find-a-way-you-murderous-so-and-so," and I guess everyone is entitled to an opinion even if it is ill informed. What I haven't met is a marine biologist, or even a marine conservationist that agrees with that opinion.
