Launcher won't start

[Packetdancer]

To add to Rhaervis' suggestions... this definitely sounds like something's gone horribly wrong in the encryption/security layer of things, so I wonder if it's possible that either the root certificates on your computer are outdated (i.e., some Windows Update didn't get installed that would have updated a certificate that's in the chain of trust for FFXIV's encryption certificate) or that your computer's clock is off by enough to invalidate something somewhere in that chain of trust? You'd think no, but I've seen both of those things affect people before when it comes to SSL or TLS failing.

[RylaBee]

PS: I just got past the "A system error has occurred: -2146697211" error,
and inputing my credentials run into second error that is often occuring:
"
Unable to complete version check.
[30410][30607][20459][10009][10019]
"

[RylaBee]

PPS: and now, after about 7 tries, I finally logged in

This is definitely caused by communication with logging servers, can't think of any other reasons than time-outs or time-sync issues with SSL.