Your focus on codeshare is misguided; codeshares do not have any effect on United earnings at all.

United-operated or United-ticketed flights' earnings are always based upon the fare. Flights that are both partner-ticketed and partner-operated earn based upon the distance, fare class, and preferred / non-preferred status. The only affect that a codeshare might have is to allow you to book a flight on a United ticket that you otherwise couldn't -- and you're correct that SQ and UA do not codeshare on their long-haul flights. Furthermore, due to carrier restrictions on the UA fares and ticketing restrictions on the SQ fare, it's virtually impossible to ticket a long-haul SQ flight on a UA ticket. The nonstop SIN-EWR flight will always credit 1000 PQP each way. That's the penalty you're going to pay for nonstop service.

I don't believe you gain anything from flying SQ JFK-FRA-SIN, because I don't think UA considers the FRA stop, but I'm willing to be corrected by someone who's tried it.

Your best option, from a comfort + earnings perspective, may be to fly NH via Tokyo.* It's about a thousand miles shorter than the SFO route -- only 500 miles longer than the nonstop -- and you have options with NH. If the fare is less than about $5000, you can book NH-operated flights via ANA and earn about 5K PQP. If the fare is more than that you can book via UA and earn full PQP.

FWIW, I find Polaris much more comfortable than SQ's A359 seat. Obviously opinions differ on that subject. The only concern that I'd have about adding an extra connection is that connecting has become perilous in the COVID era; if you don't have permission to enter Japan, you could end up with a mess if there's a delay.

* once it's operating again. Thanks Kacee.