My assumption has always been that FTR increases with experience and it's not something I typically worry about with young players. I tried finding some data online to see if that was true, but couldn't find anything. I went to NBA.com and first looked at the median FTA/100 Possessions (I don't think NBA.com has FTR) per draft year. Here's what I found:
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So the median free throw rate does appear to increase over time. It looks like there is an increase for the first 5-7 years and then it plateaus. It's possible that this increase is due to players who stick in the league are better at getting to the free throw line so that the median is increased. So I looked at the FTA/100 possessions for the 9 players in the league that were drafted in 2013 and then averaged their free throw rate by years in the league and here is what I found.
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The same trend continues, so I'm pretty comfortable at saying that free throw rate improves over experience level, but it's also probably true that for the average player it doesn't improve by that much.