I have experience of TES and friends at TAS.
The latter - based on information from friends who work there - is that they now utilize psychological profiling as a recruiting tool. I'm not quite sure what that says about the type of people they hire, but I do think it suggests that they are very specific about who they are looking to employ. Maybe they're trying to weed out any undesirables asap.
I also know from a friend who no longer works at TAS that they employ Ivy League and Oxbridge types with no teaching license. Obviously that's not the only teaching demographic there, but it provides an insight into their thinking. I presume it is due to the prestige factor associated with such an education plus many of the parents aspire to sending their children to such universities. Whilst I say all this, bear in mind it's second hand information. On the other hand, I do know quite a bit from my time at TES...
Most teachers are hired from overseas job fairs; however, I know of several teachers who were employed at TES while already working in Taiwan. These 'local hires' are fully qualified, experienced teachers who have worked at 2nd tier international schools, cram schools or EAL staff at the school. Some were traveling spouses who worked in other jobs - see above - until a position became available.
I know they were hired locally because they used to have a different contract from teachers brought in from overseas. At one time, TES British Section could employ two fully qualified teachers from England, but one was employed via the overseas route and one the local route as they already lived in Taiwan. Even though they did the same job, the local hire received no teacher package like housing and flight subsidies while the overseas hire did. It was a contentious issue which has now been rectified by providing the package to both. My point is that TES definitely employs qualified, experienced teachers who are already in Taiwan.
Importantly, don't forget that TES is three schools in one. The German section only recruits from Germany, but the British and French sections both hire locally. The French section is a damn good setup and very, very impressive. It's a bilingual program where students have two teachers to learn subjects in both French and English. Actually, I would say the French school is the most relaxed in terms of working environment.
If you really want to move, you could always sign up as a supply teacher for these schools. I'm not sure if you can do both, but I know that supply teachers can earn a decent salary at TES and probably more at TAS. More importantly, it gets your foot in the door until the next recruiting season.
I hope this helps.