I don't use Safari or any AppleWare but I disable 3rd party cookies in Chrome and I can't download files from Google Drive unless I temporarily enable it.
I poked around in dev tools, and it looks like what's happening is in serving to Safari it gives a download link that includes authentication information, but when serving to Chrome it doesn't. Perhaps because it's expecting to get third party cookies?
My guess is that they rolled out special handling for Safari when it started blocking third-party cookies by default, which will eventually make its way to all browsers?
(Disclosure: I work on ads at Google, speaking only for myself)
I think many people will agree that this is just terrible engineering and there are many ways to authenticate users who download files without the use of 3rd-party cookies. Literally every other cloud storage provider was able to figure that one out.