I worked as a TEFL teacher and ran my own language school for 10 years employing about six English teachers a year.
When going through CVs I used to start with the hand-written letter. If it was in good English without too many spelling or grammatical errors it didn’t go straight in the bin. Next came going through the CV for a degree of sorts. After that it was back to the letters and a phone call to have a chat in English and French. I don’t think anyone we employed had a TEFL qualification but several had PGCEs.
Both Madame and I had PGCEs with an EFL mention and most schools we applied to offered us jobs.
Contact the British Council, they don’t require a TEFL qualification but will look for professional experience on the CV. They pay your traveling expenses and a living wage wherever you end up.
Keep bumping the thread till Roter Stern posts.