As a former SQ PPS Club member, the SKL is not and never has been a Star Alliance Lounge. It's a separate lounge entirely, intended for First/Suites and Business Class Passengers aboard Singapore Airlines Only (and PPS club members)
Star Alliance flyers are usually asked to go to the Krisflyer Gold lounge, which is nice, but not as nice as the SKL.
I am surprised SQ is letting UA pax flying C and F in there. I wouldn't push it.
The
rules are clear:
The lounge provides an oasis of calm and respite for PPS Club members and
Singapore Airlines customers travelling in Suites, First and Business Class cabins.
Anyone flying NC, NF, C etc on UA metal is not an SQ customer