Why F404?
A stronger engine gives room to grow for heavier variants. No need to worry if engine could cause problems in a national first jet project since it is a very stable safe engine. F404 is designed for trainer jets according to GE. To minimize compressor stalls and other engine failures. Additionally, the engine was intended to respond quickly to control inputs, as novice jet pilots transitioning from propeller planes to jets often complained about the lack of responsiveness in early turbojets. Also whenever an engine purchase comes up TEI intervenes on behalf of their partner GE since they were the only company that accepted to co produce engines in Türkiye ever.
That's what happened with EJ200 vs F110 at least. Also related to that since THY is ordering 300+ planes that should also come with local engine & plane production deals. Turgut Özal was smarter than our current leaders. He made local engine and plane production part of the F-16 fighter jet purchase. PW and GE were 2 candidates, he said the company who accepts this will get the contract. Instead of just giving it away like THY is doing, same should be done to force Airbus and Boeing compete against each other.