Do I buy this on a UA ticket or LX/OS?
 

I am planning a trip from Chicago to Poland very late Nov / early Dec in business class. It will probably be outbound on OS, back to NYC on LX, and a final flight the next day on UA LGA/ORD. It looks to me as if it prices are much the same whether I purchase this through UA or LX. I do my Star mileage with United with whom I am a paltry silver, although I think this will turn me gold.

So - any issue about which airline I buy this through? And do I have any chance on the day or in advance of buying an upgrade to First on the ZRH/JFK sector?

You should purchase as UA ticket and directly from United (not Expedia and such). You will have much better and faster protection from UA vs LX or OS in case of schedule changes or refund if flight gets cancelled.

What leads you to this statement? What is better protected with UA vs LX/LH/OS?

My personal experience with UA and LX and LH during Covid cancellations and changes.

UA was quick in rebooking and refunds. UA can reissue tickets instantly while you are on the phone , while LX had to queue and can take multiple days. I was in the airport twice recently without LX ticket reissued in time, because reissues were manual and slow. I had to dispute 3 LX refunds with Amex, but several tickets refunded by UA quickly without any issues.