The whole point of empiricism, and the empirical methodology from that, is that you use observation (via your senses) and experience to gain knowledge. The more rigorous methodology that you described, and that science tries to employ, is simply to ensure a higher level of accuracy.
The thing about the empiricism is that it is essentially how the human brain works anyway, irregardless of what notions we may have about its ability to be logically and rationally coherent. From the moment we are born, certain structures and systems of neurons in the brain are activated and strengthened (or not activated and weakened) in response to the stimuli we receive from our environment.
Of course, the basic structure of the brain, and the neurons that make it up, are also, in a way, born of “observations” or “experience” of the environments and species that predated us. Those structures or uses that were highly inimical to properly survive or produce an accurate enough picture of the world to function would see themselves die out. The brain is empirical by nature, both in its constituents and how it operates.
To conclude or assert the inability of empiricism to deduce truths is to essentially question how the human mind works, which would leave all our arguments and ideas in a rut. I think by just us existing to the degree of success we currently enjoy (though, who can say for how much longer?), shows at least some truth-discerning virtue in empiricism, if truth means an accurate understanding of the world we inhabit.
Logic and rationality aren’t things that jump fully-formed and perfect into our heads either. Our ability to rationalize is honed and built up by the experiences (including, for better or worse, trusted authority figures) we have and are exposed to. This obviously leads to bumps and imperfections in the ideas we hold and the subsequent logic we employ from them.
Take, for example, children who believe in, say, Santa Claus. For them, the logical impossibility of a man able to visit billions of homes in the span of 8 hours or so does not register, since they have not had the experience and exposure to the just how many people live in the world, how large the world is, the speed one would need to travel, and the affect such speed would have on a person (plus innumerable other facts and issues). To them, it seems logically palatable to have a jolly man carrying every person’s presents visiting them all during one night.
Same can be said of people’s perceptions of falling objects. Before exposure to Galileo’s experiments (and the moon videos corroborating them), most people would feel that, intrinsically, heavier objects fall faster than light objects. While in most everyday experience, that may seem true, due to air resistance, it is not. Ironically enough, it’s people’s general observation and exposure to such a phenomena on Earth that leads their logical pathways astray. It shows the lesson that we cannot assume what we think of as logical and normal from our usual vantage point to be accurate, true, or logically consistent.
Taking into account that inherently the human brain operates mostly on emotional and visceral reactions rather than what we think of being rational and logical, it leaves a lot of chance that people have developed ideas and theories that may feel logical, but are not actually. Like the presuppositions of gods being necessary, or that they have attributes like omnipotence or anything like that.
Unfortunately, humans tend to use rationality to, instead of informing and questioning our positions, to justify and solidify them. So, you have to be careful of the fact that maybe the conjectures of all the theists about gods or what not could be more of trying to rationalize out a passed down devotion to an idea, or being a misinterpretation of the world from their limited vantage point, rather than being truth.