Correct, it doesn't directly have any bearing with Miles & More, I think you're getting different entities confused. Miles and More is the mileage program of Lufthansa, Swiss, Austrian, etc. Miles and Smiles is the mileage program of Turkish Airlines. When you earn Star Alliance Gold status with any of the Star Alliance airlines by earning the status in their mileage program (Miles & More, Miles & Smiles, Eurobonus, Asiana Club, etc.) then you get Star Alliance Gold benefits when flying any Star Alliance Airline. The benefits you're actually trying to inquire about are from Lufthansa, Swiss, and Austrian, while I think that Miles & More did a poor job explaining it to you, if you get Gold status from Turkish Airlines (Miles & Smiles) then you get Star Gold benefits when flying Lufthansa, Swiss, Austrian, and any other Star Alliance airline, including United (which you mentioned by way of US domestic flights).
The main benefits there are Lounge access, priority boarding, check-in, and baggage, extra baggage allowance, priority standby, etc. Of those benefits, the only time you'd really realize the benefit is flying domestic in the US where you'd get access to the United Clubs. Additionally when you fly any of the airlines in the LH group you'll have access to the Senator Lounges vs the Business Lounges (flying business class), the Senator lounges are a bit nicer usually. Aside from that, you're already going to have the other benefits by way of your First or Business ticket. Something you may consider to counter the US lounge access if you won't make status is to just outright buy a United Club annual membership for the times you're traveling in the US (however it won't help if you're not flying UA, same with Star Alliance Gold, you've gotta be flying UA on US domestic flights to benefit).
I will warn regarding TK's program if you go that route is that the miles are sometimes difficult to use. I believe they still require you to go to a TK ticketing office to have award tickets issued, which is a quite onerous requirement considering most carriers do it all online or the phone.
Regarding your point of earning a lot of miles in Alaska's program, it's correct but it's also a quirky program that may or may not have awards that are useful for you. Definitely take time to read up on it before going that route. It's primarily built for US based flyers and doesn't have the same flexibility that most programs do in the alliance carriers. (Just one example is that they can book TN (Air Tahiti Nui) from the US to Tahiti but not Japan to Tahiti. Lots and lots of examples of regional restrictions like that when trying to book their partners, it's absolutely not the entire partner network.).
How much more Star Alliance flying will you do this year? Do you live in a city with a TK booking office? Also what's the mix of SQ and LX flights on your itinerary? Is it LX from Europe to Singapore or are you also using SQ metal on those flights?