JH said in 2014 when he was still VP of Flight that letters of rec weren't looked at until after f2f interviews, and even then weren't looked at 90% of the time. The only case where they are used is if the board is on the fence about a potential new hire. Then and only then would letters of rec be referenced.
Hopefully they weigh them more heavily now, but I wouldn't be the least bit surprised if the process in 2014 is still the process today.