I was hoping to post yesterday when I encountered this error as well, but it seemed the Lodestone didn't recognize that I had played for a few minutes the morning of, and wouldn't let me log in.

Anyways, I had encountered this as well. I will post the facts of my incident:

-Like mentioned, my PS4 was on rest mode when the power went out briefly.
-As usual, the PS4 did it's recovery thing and scolded me to not "turn off the power" bc it can corrupt data. (As if it thinks Im just willy-nilly pulling the power cable out of the wall, or regularly flipping my bedroom breaker switch)
-I did not have SB on standby mode. I had physically exited the game via the menu, and had closed application before putting my PS4 into standby mode.
-When I discovered that I had to reinstall all 22GB worth of patches, I went over to my storage and saw that there were 69GB worth of data on the HDD for SB.
-I attempted to do some hotfixes: restarting PS4, turning PS4 off/on, and reinserting the FFXIV Online disk. None worked.
-I pressed Options > Delete on the PS4 menu over the SB application and reinserted FFXIV Online disk to reinstall. Checked storage and found that SB was now only 11GB big
-After taking 9 hours to download the patches, I rechecked the storage, and it was back up to 69GB.....

So I want to assume that the patch data was indeed all there. All 22GB of it when it prompted me to redownload. However, I am guessing that due to the power outage, it somehow corrupted some file that lets the client know that we have all 22GB already installed.

For now, Im taking the safe route by
1) Exiting game via FF's in-game menu
2) Closing application once the client shows the log in screen
3) Turning PS4 off (not on safe mode)