You would get into the MOE, no problem at all.
Only about 5% of Singapore would be able to pass a TEFL.
Not even English graduates are allowed to teach English to foreigners without the TEFL in most European Countries....Though English teachers would be peeved about that.
The TEFL is a specialist niche area of teaching for foreigners, that also concentrate on sound pronunciation, your income would be much more in the private sector though.
I personally think you have no problem at all, to find work any where in the world with the TEFL. I was thinking of taking it myself years ago for the same reasons, its a specialist skill.
Teaching is not a specialist skill and that's what you must emphasise, though MOM should be aware of it.
Very few Singaporeans could qualify.