You err in three different ways:
1. The validity of evolution in no way depends on the question of how life first arose, just as the validity of meteorology (the study of weather) depends in no way on where the air came from originally. Whether the atmosphere was poofed into existence by God, coalesced out of a solar nebula, was belched out by volcanos, or was poured into place by gigantic alien space ships, the properties and behavior of the atmosphere remain the same and depend only on its current composition, and meteorology is just as valid no matter what the source of the atmosphere. And it's the same for evolution -- evolution concerns the history of life on Earth, and the properties of living things and how they change over time due to the relevant processes, once life existed, no matter *where* it "came from". When you drive a car, does the physics of energy and momentum by which it works -- and you drive -- depend upon the source of the metal it's made from? No. Does the operation of an internal combustion engine depend upon the source of the gasoline it burns -- or only upon its chemical composition? *Wherever* life came from, evolution accurately describes what has happened to it since.
In fact, anyone familiar with evolutionary biology would understand that it is *isolated* from whatever processes originally formed life. Evolution can only take place when *reproduction* exists. Since reproduction obviously was not occurring before the first things which we might accurately label as "living", the original formation of life *necessarily* occurred by processes other than evolution. The nature of those processes, whatever they might have been, are irrelevant to evolution itself, and evolution is irrelevant to the original formation of life. It is either ignorant or dishonest to try to pin the validity of evolution to whether or not anyone can demonstrate how the NON-evolutionary origin of life occurred. It's like blaming the Bush administration for events which occurred before he took office. Abiogenesis was pre-evolution, and occurred by different processes. Evolution stands or falls on its own, it is not dependent upon any theory (or lack of theory) of life's ultimate origins. Even if God dropped the earliest life forms onto the planet, evolution *still* demonstrably shaped them thereafter.
2. Evolutionists *have* "checked the accuracy of their belief" in a mind-boggling number of independent methods, countless times over the past 100+ years. It has passed these tests with flying colors, and has survived all potential tests of falsification.
3. Contrary to your unfounded presumption, research into abiogenesis (the origins of life) have been extremely fruitful. The picture is still incomplete of course (but then, so is the atomic theory of matter and every other scientific field of study), but there have been a vast number of findings which, while not conclusive, *very* strongly indicate that the "life arose naturally" hypothesis is on the right track. Your apparent notion that science "cannot demonstrate" or support any aspect of this paradigm is extremely mistaken. The hypothesis of abiogenesis makes a huge number of predictions about what we should find when we look at the evidence, and in the many ways which we have to date been able to test these predictions, they have been confirmed.
"Evolutionists *have* "checked the accuracy of their belief" in a mind-boggling number of independent methods, countless times over the past 100+ years. It has passed these tests with flying colors, and has survived all potential tests of falsification."
This is untrue. However, when it doesn't pass, it is simply classified as "an unsolved problem in theoretical biology" or similar language.