I have been arguing this point since long before the Extra Credits episode. In fact, I was quite amazed when it came out, because they covered almost all of my favourite points.
I've stopped getting into these arguments however, because it seems to be a black-or-white issue with people's understanding. I find it boils down to whether or not you correctly understand the term used, "faith", which in this philosophical context has a very specific meaning: accepting unprovable axiomatic statements as true. This is the fundamental basis of all positivism.
Some people who like science (and I say this in place of 'scientists', because most actual scientists have studied Philosophy of Science in college and already know this) object to the term faith, because they associate it with religion. However, they don't really understand the implications of saying "I have no faith in any axioms." This is essentially the definition of ontological nihilism: nothing is true or logical.
These same people follow this up with "I don't need faith, I rely on evidence to prove my axioms," but this simply demonstrates a major lack of understanding of how science and logic actually works. Every mathematical or scientific proof begins with an axiom, i.e. the results of another proof. There is no such thing as an original proof: if your axiom is "true because it is true", then it is known as a tautology and is considered logically unsound. No matter what you offer to prove another statement, another scientist can keep saying "now prove that." Eventually, you both have to agree on something that is taken for granted as true, rather than enter into an infinite loop of trying to prove each successive axiom. This is technically an act of faith.
It may be a much slighter act of faith than say, believing in the bible. But that's just a matter of scale. I don't think anyone is arguing that the scientific method requires as much faith as religion. Only that fundamentally, the two systems of belief are configured much the same, just with different axioms.
This is my understanding of things, as a current MSc student who has studied philosophy of science in university.
Take from this what you will. I just wrote this post in the hopes that it will help some genuinely science-minded people to overcome their inherent defensiveness (I was the same way when the topic was first broached in uni) and consider this logically.