I have not followed closely but I thought the last barriers to the F-16 deal had been removed and a contract had been signed.You should.... avoid. . At least in the defense sector, you should give them a wide berth. It's fine the way it is, the F-16 Block 70 including the upgrade kits will be the last planes from the US.
If we don't get them, we shouldn't waste any more energy trying to convince them, just convert the remaining F-16 Block 40/50s into Özgur II, the equivalent of the Block 70 Viper.
The delivery of the F-110 engines only came about because General Electric is a minority shareholder in TEI and is therefore forced to fulfill contracts that have already been concluded.
If the US government had torpedoed the deal, the Russians or Chinese would have agreed to supply the engines together with a technology transfer as in the S-400 deal.
Both have asked to help Turkey with the engines if their wonderful allies refuse.
That would have been the final straw for Turkey's NATO membership, which would then only be on paper and worth no more than a scrap of paper from Turkey's point of view. It would still be a NATO member, but would not provide any assistance to the wrong allies.
If Russia or China would be happy to help with engines, that is good. What is not so good is that Russian engines are not as good as US, French and British engines and Chinese engines even less. I imagine that Russia would welcome making money out of Turkiye. Who else is interested in Russian help with engines or in buying Russian engines to power their own designs?
Out of curiosity, why does Turkiye want to stay in NATO? Who would attack Turkiye so that it needed to call on help from its NATO allies?
PS Why buy US upgrade kits if you have your own?