Remember, almost half of the classes at HYS have <~171 LSAT scores (and/or <3.8 GPAs). So even if there are roughly 900 folks applying to HLS each year who have >171 LSAT and >3.8 GPAs, not all of them are getting in. I think it's fair to say that likely admission (e.g., "you have a good shot!") to HYS is somewhat of a numbers game. Thus, if you have a 175 LSAT with a 3.85 GPA, you're likely to get into at least one of HYS. Application to HYS can be a numbers game if you have the numbers--there are few enough applicants with both LSATs and GPAs above HYS medians that many if not most of them get into one of HYS.
But these aren't the folks who make up the majority of the class at HYS. And for those closer to the medians, it becomes much more about other things. Thus, a fairly substantial number of folks with 169 or 170 LSATs get into HYS, and a fairly substantial number of folks with 3.4 or 3.5 GPAs do as well. And a fairly substantial number of folks with 172/3.7s don't get into any of HYS.
If I had to bet, I'd bet that of the roughly 6,000 applicants to Harvard, around 1,000 have solidly above-median scores, around 3,000 have scores near the median, and around 2,000 have either an LSAT or a GPA score (or both) well below the median. And, I'd bet that admissions rates are something like 500/1,000 (50%) for the folks with above median LSATs and GPAs, 300/3,000 (10%) for folks near the medians, and 100/2,000 (5%) for folks notably below median. I'd also guess that of the folks who get in with below-median scores, most of them (90/100?) are splitters and/or URMs.*
If you buy the above numbers, it becomes clear how softs do play a fairly big role in even Harvard admissions. My sense is that they play an even bigger role in YS admissions.
*I rounded a lot because I felt stupid guessing something specific, like that 700 applicants have above median scores (which is probably closer to what my actual guess would be).