Gergely, I have a similar AMD motherboard as you, and when I first tried playing with flashrom at the end of August I installed flashrom on FreeBSD. I got almost the same message as you:
"FCH device found but SMBus revision 0x59 does not match known values."
(Except mine said revision 0x61)
Well, I am retired and messing with computers is my hobby, so I thought maybe trying a Linux would work better with flashrom, and installed Ubuntu. I still got the same message, and no real answer until Rene' Rebe posted this email on Oct 12:
What I realized was that I needed to download the source, modify a line to include my particular spi chip, compile and install flashrom myself. I looked at the documentation, asked questions, and IT WORKED.
And, besides being able to read my flash chip with flashrom, I have been able to use it to write a stock bios update from MSI on my x570 motherboard, and IT WORKED. The first place to look is here:
So look at Rene's email & the flashrom download & install info, and ask more questions. Most everyone here is very willing to help.
And, there is a fix in the works on Gerrit by the flashrom developers, but it's not yet released. Unfortunately it doesn't cover your x370 motherboard, but I'm going to try to get that changed to include 0x51 too.
You might enjoy the #flashrom & #coreboot channels on Freenode IRC.
Best of luck,
Clay