Hays was chairman of the RNC, campaign manager for republican president Warren G. Harding, and postmaster general under Harding. Not just any Republican politician, but a leader in the GOP.
Personally, it seems to me to be a code developed at a time of differing cultural values, that likely would have been supported by any mainstream politician.
There's no way you can meaningfully associate the modern political parties with pretty much anything of the 1930s, because aside from the names, little is the same about the parties. You could blame the Hays Code on pressure from Catholic social conservatives, and observe that that is now a GOP constituency, but how meaningful that is depends very much on what you are trying to get to from it.
> Catholics have been solidly democrat until 2016.
No, they haven't; they’ve been split roughly evenly between the parties for much longer (this data only goes back to 1992, and shows it true the whole time):
https://www.pewresearch.org/politics/2016/09/13/2-party-affi...