I think "she" is a better approximate sound than "see", if we go with the Putonghua phonology
however, "see" might be more legitimate if we consider the jiantuan distinction in historical phonology (old national pronunciation) and many contemporary dialects let me explain https://twitter.com/mattsheehan88/status/1314025403317583872

The Putonghua as we speak now abolished the jiantuan or sharp-round distinction, which is preserved in many dialects and operas (e.g. in Peking opera)
In PTH 剑 sword and 箭 arrow are both jiàn.
But traditionally 剑 is jiàn, 箭 is ziàn.
This is called jiantuan distinction

Other examples:
心 is sin instead of xin in Peking opera and other dialects that preserve the distinction
星 is sing instead of xing.
and... 习 is syi and 进 is zin. Both are "sharp" 尖音 rather than "round" sound 团音.
Therefore "see" is the traditional pronunciation!

the consonants in 幸 xìng (团音 round sound) and 星 xīng (尖音 sharp sound, used to be "sing" in Mandarin) must be very different in older times and we can see that from other Sinitic lans.
in Cantonese the consonant in 幸 is "h"; 星 is "s".
In Wu 幸 is "yin" and 星 is "sin"