That's the ideal case, but in practice much of it is directly derived from natural gas instead of electrolysis
In 2022 less than 1% of hydrogen production was low-carbon.[1] Fossil fuels are the dominant source of hydrogen, for example by steam reforming of natural gas.[2]
It's not zero evidence, it's just not 100% certain. There have had known H5N1 infections in other mammals consuming raw milk that spilled. Many of them have died from it. Additionally, the testing that has happened shows that 1 in 5 US dairy samples are positive for H5N1 so it's prevelence is rather high making there be a very real risk