I am very committed to LibreSSL and am very annoyed at the OpenSSL development process, which seems to be more about getting money from the FIPS consulting than anything else these days.
If bitcoin-qt didn't work with LibreSSL (as it didn't until recently) then I would have no choice, but since bitcoin-qt does work with LibreSSL then I have a choice, and that choice is to ditch OpenSSL wherever possible.
That's why I run a CentOS 7 yum repository based on LibreSSL with lots of different software rebuilt against LibreSSL. I don't trust the OpenSSL library to be secure.
You're trying to convert a missionary here, I'm also inclined toward LibreSSL after the problems with OpenSSL (and well done for putting it together incidentally, it must have been challenging whichever way you approached it).
But not for this one purpose, it's part of the consensus driven code base. With crypto-driven consensus, correctness is not important if there's a practical reason to stick with incorrectness in respect of long-term development goals (like the Core developed in-house libsecp256k1 library mentioned by knightdk, that's been several years in the making now and only just about to see a release)