I'm not sure you can state this as a fact. Ecosystems and ecosystem services are so complicated that it is pretty much impossible to say how bad the disappearance of a group would be for the ecosystem. The safe option in my opinion is to try to keep everything extant, the idea being that if we prevent extinctions we don't have to worry about unknown ramifications. Whether or not you like snakes, you should try and keep them around since they're probably doing something important. They are, after all, extremely abundant in this country and without them where will all that energy go?
Unless it happens, which it won't, it's hypothetical and thus we can't prove it, but as a qualified ecologist I have no doubt there would be a fairly minimal ecological impact. This doesn't for a moment mean I want it to happen or I would encourage it. I like snakes for their own sake. If they vanished I'd think it was a great shame, not because the world would end, but because those snakes would be gone. For everyone else however, it would probably be considered a good thing. Without any reasonable doubt, we could wipe out a lot of the most deadly snakes and their niches would then be filled by less venomous snakes. This definitely would have virtually no impact on the ecosystem. Again, I don't want it to happen, but if we were to be honest with the general public they would want to take this option.
Anyway, in reality, if we all did our best to kill as many snakes as we could (as opposed to the current situation where only most people do that, which is pretty much exactly the same thing) we'd still be unable to wipe them out. That's why they haven't already been wiped out. Snakes are pretty difficult to find in the wilderness (or even in urban situations) so while a lot of individuals get shovelled, as species, direct human attacks post basically no danger in almost all cases.
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In many countries were they exist (including Australia) snakes are one top end predators that control the numbers of not only rodents but also species of amphibians, other species of reptiles, mammals, birds and fish and if they were to become extinct it would have a major impact on the ecological balance of many environments. It would dramatically increase the numbers of these species causing a domino effect on the consumption of their own food (not just animals but also vegetation) which would in turn would lead to a major collapse in all population. In addition animals that prey on snakes would have to adapt to prey on other animals to maintain their survival which would drive down the populations of those species. And if the new prey items are reduced or the predator of snakes fails to adapt then this will cause a further collapse.
This myth is even more incorrect than it is common. They're not usually top end predators, especially in Australia. They're always predators, but rarely top end. But either way, they don't play an important role in controlling rodents. It wouldn't cause some crazy domino effect, it would just cause a very minor increase in alternative predator numbers (snakes aren't the only things which eat frogs, rodents, etc, and if you take away snakes, the reduced competition simply allows the other predators to increase in numbers and the system is in balance).
Laymen often believe the myth you are talking about, but actual ecologists understand that while introducing new species is sometimes catastrophic, taking away mid level members of the food web generally has little effect. Many studies have been done to show this, and I've been involved in such studies myself, including reptile examples in Australia.
Incidentally, removing top end predators is the least impact of anything. The stuff at the top can be taken away without it doing much to what's below it. If you remove the foundation, that's what causes the biggest problems, but in the case of snakes in most cases, Australia being a good example, it just wouldn't do much.
As you may know, snakes don't actually eat very much, so they aren't especially good vermin controllers. Things like owls/tawny frogmouths eat far more and are much better at it. Even feral pest species do far, far more for vermin rodent control than snakes.
I know snake enthusiasts desperately *want* to believe the 'snakes are critical vermin controllers' myth, and that makes them repeat the story so many times they generally firmly believe it to be gospel, but in reality it's not true.