I did a TEFL course because I actually wanted some idea of what I was supposed to be doing before I entered a classroom. I had no experience of teaching; therefore, I didn't think anyone would employ me. I've had three jobs in Thailand and not once did anyone ask to see the certificate. They only cared about my degree for work permit purposes. Even after I did my course, I was still completely useless. I look back at some of the things that I did and I'm embarrassed. But hey, you gotta start somewhere.
I see many jobs on ajarn now require a degree in education. That's fair enough, but often they say that no experience is required or at least a year. Sorry, but even with a degree in education, you're still going to be fairly useless with zero or one year's experience. I've worked with newbies with education degrees and even they'll admit they need guidance and to gain experience.
Experienced teachers who actually try are worth their weight in gold. You can only learn so much from a book and then you really need to just get your hands dirty. Also, a degree in education is very general and usually more accustomed to the West. Certainly not countries like Thailand (unless you're working in an international school).
For your bog standard teaching jobs, I'd choose experience over qualifications every time. For international schools, of course an education degree. As for some kind of TEFL, I'd say do it for the experience. But when starting out, know that you'll get the same pay as someone starting with no TEFL. Like most people I worked with.