The "Request Desktop website" option still exists in iOS 13 for iPhone as it has in previous iOS versions, though it has been moved to a different location, under the Font / Text Size icon in the address bar.
With that said, its entirely up to the website how they respond to that. Many manage their mobile website based on screen dimensions so requesting the Desktop version is not going to change that if there is no other version to offer.
I know because my website does exactly that. It does not offer a desktop view on iPhone, even when requested because the mobile site is based on screen width and that does not change by requesting the desktop site. in other words, the desktop and mobile websites are one and the same only adjusted for width.
However if I go to say Amazon.com, and request the desktop version, it responds correctly and shows me the full website rather than the mobile version.