For your needs
- 4-8 round-trip flights FRA-OTP, half of them with two teenagers (over 12).
This means you will mostly take LH or possibly OS flights. MM group is very stingy at mileage accrual to TK.
- Our family's preferred vacation destinations are the US/Mexico.
No TK has great advantage if you are based in Germany and flying Asia. TK easily has 5 flights to major Germany cities (FRA/MUC/STR...) every day, which means flexible connecting time in IST. For US/Mexico I think LH and UA is much better.
- Travel to the UK 1-5 times per year.
Again this means mostly LH/BA flights. BA is not TK partner. LH intra-EU flight and empty middle seat business... you know what I mean. TK is great if you fly a lot EU-IST routes, their business service is certainly the best in Europe.
- Moderate vacation travel within Europe, 1-3 times per year.
OK but SA gold no longer has 1st free bag in LH short haul flights.
-Long haul after qualifications mostly with award flights.
No, TK award ticket is painful.

I assume UA is a better option if you want to quit M&M. LH miles accrual to UA is not to bad. UA offers more options and advantages for your need to US/Mexico. UA award ticket system is one of the best in SA.